Elastic Waves in Layered Media

McGRAW-HILL SERIESIN THE GEOLOGICAL SCIENCES UNIVERSITY OF FLORIDA LIBRARIES ENGINEERING SCIENCES UIBRARY Digitized by the Internet Archive in 2010 with funding from Lyrasis IVIembers and Sloan Foundation http://www.archive.org/details/elasticwavesinlaOOewin McGraw-Hill Series in the Geological Sciences Consulting Editor Robert R. Shuock, ELASTIC WAVES IN LAYERED MEDIA McGraw-Hill Series in the Geological Sciences Consulting Editor Robert R. Shrock, DE Sitter • Structural Geology • EwiNG, Jardetzky, and Press Elastic Waves in Layered Media Grim • Clay Mineralogy • Heinrich Microscopic Petrography • Shrock and Twenhofel Principles of Invertebrate Paleontology Elastic in Waves Layered Media W. Director, MAURICE EWING Lamont Geological Observatory Professor of Geology, Columbia University WENCESLAS S. JARDETZKY Associate Professor of Mechanics, Manhattan College Research Associate, Lamont Geological Observatory FRANK PRESS Professor of Geophysics, California Institute of Technology Lamont Geological Observatory Contribution No. 189 McGRAW-HILL BOOK COMPANY, INC. New York Toronto 1957 London ELASTIC WAVES IN LAYERED MEDIA 1957 by the McGraw-Hill Book Company, Inc. Printed in the Copyright United States of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card © Number 56-7558 The interplay of the two methods guided the research program which led to this book. Samuel Katz. Dr. the Bureau of Ships. Dr. and Stefan Mueller kindly read the manuscript and made helpful suggestions. and in many problems of and electromagnetism. and it has been retained whenever possible. Peter Gottheb. sentation of problems concerning elastic-wave propagation useful in other fields. Franklyn Levin. which of problems. and model seismology carried on by the group connected with Lamont Geological Observatory of Columbia University. Cambridge Research Center. Laughton. underwater sound. governed the selection For many years. and the Office of Naval Research for support of the program of research on elastic-wave propagation at the Lamont Geological Observatory. The experimental viewpoint are scattered in many journals. The methods and results of the theory of wave propagation in layered media are important in acoustics in geophysical prospecting.PREFACE This work is the outgrowth of a plan to make a uniform presentation of the investigations on earthquake seismology. It is hoped that a systematic pre- seismology. and SOFAR sound propagation are a few examples of topics in which the theoretical and practical investigations benefited each other. and shock waves are very close to the methods used in this book. Few workers in this field could become familiar with all the past investigations. flexural waves in ice. to a large extent. water waves. An effort was made to compile a comprehensive and systematic bibliography of the world literature for the main topics discussed. we had to reduce them to a few brief references. Dr. Although the mathematical discussions of electromagnetic waves. may now be has. Observations of surface waves from explosions and earthquakes. A. The scope was subsequently enlarged to cover a particular selection of related problems. Many of the methods which have been used in seismological problems were originally developed in studies on electromagnetic waves. We are very grateful to the Air Force Maurice Ewixg Wenceslas Jardetzky Frank Press . research in seismology has been characterized by separation of the experimental and theoretical methods. . Special Forms Used 20 24 24 26 28 Homogeneous and Isotropic Half Space 2-1. Elastic Media Isotropic Elastic Solid Ideal Fluid 1-3. 1-5. Media Boundary Conditions Reduction to 1-6. Integral Solutions for a Line Source Surface Source Internal Source 2-4. Imperfectly Elastic 1-4. Reflection of Plane Waves at a Free Surface Incident Incident P Waves S Waves Energy Partition of Reflected 2-2. Other Investigations Time Variation 64 vii . Free Rayleigh Reflection at Critical Angles 28 29 31 Waves 2-3. Integral Solutions for a Point Source Surface Source Internal Source 2-5. Solutions of the Wave Equations Wave Equation Plane Waves Spherical 5 5 5 6 7 7 7 10 10 12 16 17 18 Waves Poisson and Kirchhoff Solutions General Solution of Wave Equation Some References 2.CONTENTS Preface List of 1. Generalization for an Arbitrarj^ 2-7. Evaluation of Integral Solutions Application of Contour Integration Residues 34 34 36 38 38 42 44 44 47 Branch Line Integrals: Line Source Application of the 49 Descent Method of Steepest 59 61 2-6. v Symbols xi 1 1 Fundamental Equations and Solutions 1-1. Equations of Motion The Equation of Continuity 1-2. Two Semi-infinite Media in Contact of 74 74 74 3-1. A Layered Half Space 4-1.Vlll CONTENTS Study of 2-8. Other Investigations References 115 121 124 124 4. Solid Layer over Solid Half Space Rayleigh Waves: General Discussion Propagation of Rayleigh Waves across Continents Ground Roll Theoretical Rayleigh-wave Dispersion Curves Love Waves: General Discussion Love Waves across Continents Love Waves across Oceans Lg and Rg Waves Other Investigations 4-6. Air-coupled Rayleigh 4-8. Three-layered Half Space Oceanic Rayleigh Waves with Layered Substratum Love Waves 4^7. Reflection of a Pulse Incident 3-3. General Equations for an n-Layered Elastic Half Space 4-2. Reflection and Refraction Rigid Boundary General Equations Plane Waves at an Interface Liquid-Liquid Interface Liquid-Solid Interface Solid-Solid Interface 3-2. Traveling Disturbance 2-9. Two-layered Liquid Half Space Discussion of Solutions Generalization for a Pulse 4-3. Experimental Lamb's Problem • • • 67 69 71 References 3. Liquid Layer on a Solid 126 137 142 151 Bottom 156 157 Compressional-wave Source in the Solid Substratum Suboceanic Rayleigh Waves: First Mode Compressional-wave Source in the Liquid Layer 166 174 184 185 189 189 196 200 Leaking Modes Some Aspects of Microseisms 4-5. Further Waves Remarks on Waves Generated 113 3-5. Propagation in beyond the Critical Angle Two Semi-infinite Media: Point Source 76 78 79 83 93 93 Two Liquids Solids 94 105 107 Ill at an Interface Fluid and Solid Half Spaces Two Stoneley 3-4. Waves of Air-coupled Ground Roll Remarks Concerning the Problem an n-Layered Half Space References 204 205 213 216 219 222 224 225 227 230 236 238 245 . Three-layered Liquid Half Space 4-4. in Media with Variable Velocity 328 Propagation in Heterogeneous Isotropic Media 7-2. Cylindrical 6-6. Love Waves in Heterogeneous Isotropic Media Homogeneous Layer over Heterogeneous Half Space: Matuzawa's Case Sato's Case Jeffreys' Case Meissner's Case Wave 328 330 335 341 341 . Liquid Cylinder in 6-8.CONTENTS 5. SH Waves SV and P Waves Crary Waves Flexural 6-4. Effect of Gravity on Surface Waves in Ilayloifrh Viscosity 255 255 257 257 260 263 263 265 266 269 272 272 276 277 278 281 5-1. Effect of Curvature on Surface Waves Cylindrical Curvature Sperical Curvature 5-4. ano General ]'](|ualions 5-2. Cylindrical an Infinite Solid an Elastic Medium Tube References 7. CunvA'vuim. Propagation in a Gravitating Compressible Planet The Effect of Internal Friction Voigt Solid Maxwell Solid [ Internal Friction in Earth Materials References 6. Floating Ice Sheet . 343 344 346 347 . General Solutions for a Spherical Body Wave 5-5. Gravity Terms Waves: Incompressil)lo Half Space . Cylindrical 281 283 285 286 286 288 288 292 293 293 293 293 295 299 301 Waves from an Impulsive Source Rod in a Vacuum Longitudinal Vibrations Torsional and Flexural Vibrations 6-5. Plates and Cylinders 6-1. . . Sound Propagation in a Fluid Half Space SOFAR Propagation The T Phase 7-3. Plate in a Liquid Symmetric Vibrations Antisymmetric Vibrations Other Investigations 6-3. Plate in a Vacuum Symmetric Vibrations (Mi) Antisymmetric Vibrations (M2) Interpretation in Terms of P and Impulsive Sources SV Waves Other Investigations 6-2. . . Gravitating and Compressible Liquid Layer over Solid Half Space 5-3. ix Tub Effects of Gravity. Cylindrical Rod Hole in a in Liquid 6-7. 305 306 311 314 314 317 319 323 Wave Propagation 7-1. Mantle Rayleigh Waves 7-5. Rayleigh CONTENTS Waves in Heterogeneous Isotropic Media '. Method of Steepest Descent 365 369 ^ .X 7-4.• • • 349 353 ^^^ Appendix A. References Appendix B. Aeolotropic and Other Media References . Rayleigh's Principle References Index . Group velocity w) Velocity X. ^('Ai. V. Hydrostatic pressure w T s(m. ^2. Pzt (P components Principal value (of an integral) p q.) CO Rotation Angular frequency xi . • • • . 'As) Displacement potentials Velocity potential ^ a(i2„ n„ 12. e^y. V. • • • . e. n p Lame constants Density Poisson's ratio or an angle a <P.LIST OF SYMBOLS Cr c Velocity of Rayleigh waves E Gxz. Pxv. I Modified Bessel function of the second kind of the order n Wave Stress length Pxx. id) Displacement components in cyhndrical coordinates Displacement Period U v(w.^ e / f H„^\ Hl^^ In t Phase velocity Young's modulus Strain components Angle of emergence Frequency or constant of gravitation Angle of incidence for shear waves Hankel functions of the order n Modified Bessel function of the first kind Angle of incidence Bessel functions of the order of the order n J„ n k Wave number Coefficient of incompressibility k 5C„ lo. Z a /? Body forces 7 e Compressional-wave velocity Shear-wave velocity Parameter Phase shift Cubical dilatation or an angle Critical angle 6 dcr K Root of the Rayleigh equation or parameter X. Y. . When a deformable body undergoes a change in configuration due to the application of a system of forces. and of constant thickness. w. We begin with a brief media and a derivation of the more detailed treatment may be found in reference books. isotropic. y + A^/. e. A being. we may write the displacement components (1-1) in the form + V + IV + u fNumerals of the chapter.Az) + e. higher-order terms can be neglected...g.r + e^. z and n. Then. each layer being continuous. (0„A2 - fi.. its displacement components can be given by a Taylor expansion du Ay dy dv_ in the form u + + + du Ax dx dv + + + + + du A2 dz dv + + -{- V dx Ax dy Ay Ay Az dz (1-1) w dw Aa: dx + dw dy dw dz Az For the small strains associated with elastic waves. the components of displacement outline of the theory of motion in elastic equations of motion. cyclic change of letters x. u. Equations of Motion. z v. (e„..A. The problems we shall consider concern the propagation of elastic disturbances in layered media. Within the body.Ax) + + + (e„Aa.A^) {Âx (nÂy nÂz) n.Az) numbered (1-3) in brackets in the text correspond to the references at the end ... any point P with space-fixed rectangular coordinates {x. + A2). z) is then displaced to a new position. y. Sommerfeld [57].A.T (e. the body is said to be strained.^z) + e. respectively. y.^y + e^.Ay + e.f CHAPTER 1 FUNDAMENTAL EQUATIONS AND SOLUTIONS 1-1.Ay + e. w. If Q is a neighboring point {x + ^x. introducing the expressions 1 (dv du\ 1 (dv du\ and others obtained by the respectively. v. is It is also shown in the theory of elasticity that there a particular set of orthogonal axes of strain vanish.2 ELASTIC WAVES IN LAYERED MEDIA first terms of these expressions are the components of displacement be shown that the terms in the first parentheses correspond to a pure rotation of a vokrnie element and that the terms in the second parentheses are associated with deformation or strain of the of the point P. e. As A*S goes to zero. 1-1). respectively. (1-5) neglecting higher-order terms.. since e^y = Cy^ • • ' .S to the area A<S is finite and defines the . Thus the deformation at any point may be specified by three mutually perpendicular extensions. the result holds for eyy.^ and the other three are the shear components of strain. defined as the limit approached initial of increase in volume to the by the ratio volume when the dimensions A. Although the principal extensions are used in the derivation of (1-5). A^. zx planes.^ any cartesian system because of the in variance of the sum.r lim + 5 e^^ Ax)(A?/ + e^y . zero. which may be shown to be equal to half the angular changes in the xy. Av)(A2 . of an originally orthogonal volume element. equivalent to a resultant force or traction R upon the element and a couple C (Fig. in general. The array 6x1 ^VX "xy "xz ^VV ^V3 \^~^) represents the symmetrical strain tensor at P. P &ZZ for which the shear components These axes of known Qyy.) — Ax A?/ A2 • A.r \y A2: = e„ + e. = -+-+e.„ + 6.^. y. It can The element. i\z approach . z axes. as the principal axes of strain. the limit of the ratio of traction upon A. through are ^xx. Cy^. yz. The cubical dilatation Q. is (A. The corresponding values are the principal extensions which completely determine the deformation at P. Ay.. The three components _ du _ dv _ dw x. e. represent simple extensions parallel to the expressions e^y. Forces acting on an element of area A>S separating two small portions of a body are. 7 + e. It is also known that the sum ^zz Is Independent of the choice of the orthogonal coordinate ^xx + ^yy + system..^. z axes. py. This may be proved by considering the equihbrium of a small volume element within the medium with sides of length A. it is necessary to may be reduced to comtractions all these However. involving an additional dimension For a complete specification of the stress of length. parallel to the x. 1-1. of area A<S. The ratio of the couple to Â Fig. .t. may be neglected.u Pvy Px.u Pzz where the first subscripts represent a coordinate axis normal to a given plane and the second subscripts represent the axis to which the traction is parallel.. This gives nine elements of stress (see Fig. As. since the corstresses p^„. P. 1-1) Vr. Pu2 (1-6) P. The array (1-6) is a symmetrical tensor. through the point./.. Traction Vvvt Vyzj R ^^^ Pyx in plane and couple C acting on element normal to y axis.. Stress components ponent tractions across planes parallel to the coordinate planes. A. responding forces intersect the axes through the center of mass of the • . Pui p. Across each of these planes the tractions may be resolved into three components parallel to the axes.FUNDAMENTAL EQUATIONS AND SOLUTIONS stress. 3 A*S'. traction at P acting upon all planes passing give the at P. y. Moments about axes through the center of mass arise from tractions corresponding to Moments of normal stresses vanish. AS. (again neglecting higher- order terms) . To (1-6) derive the equations of motion we consider the tractions across the surfaces of a volume element corresponding to the stress components and the body forces X. z axes. 1-2). AS^ are the areas respectively. p^y = py^. etc. It follows that the x of the faces normal to the x. AS. component of force resulting from all . Then the stress at a point is completely specified by the principal stresses p^^. the volume element (Fig.. Z which are proportional to the mass in Fig. that the shear The equilibrium conditions require. Stress components in the faces ASz of a volume element. ASy. components of stress be equal in pairs. p^. where AS^. z axes. three mutuallyperpendicular axes of principal stress may be found with respect to which the shear components of stress vanish. Y. y. is e. corresponding to these axes.g. y. When the tractions are considered.4 ELASTIC WAVES IN LAYERED MEDIA infinitesimal element and moments of body forces are small quantities of higher order than those of stresses. therefore. the X component of the resultant force acting on an element. produced by stresses in the faces normal to the x. 1-2.^- + ^ ^^' AS. pyy. As was the case for the shear components of strain. ... This equation expresses the condition that the mass of a given portion of matter is conserved. it is assumed that each of the six the components of strain... y. The loss of mass during the same time is — (dp/dt) At At.. The total outflow of mass from the elementary volume At during the time At is div pv At At. 1-2. Isotropic Elastic Solid. using . In the generalized form of Hooke's law.. z axes are M. dp. w. and the stress-strain relations may be written in the foUowmg manner. dz dp. dp... appear in the stress-strain relations. dx dy dp. Equating these last two expressions gives 1^ + is div pv = (1-8) Another form of this equation ^+ where the operation p div V = (1-9) 1 and d/dt is = 1+ v. 17"^^ dp. ^^"^^ '1?= '^ ^^-^^-^^ In these expressions.. whose components parallel to the x. d\i dt „ dp. Elastic Media. isotropic body. V.grad (1-10) represents the "total or material" rate of change following the motion the local rate of change. where v is the velocity. p is the density of the medium. dp. the components of stress is a linear function of all and in the general case 36 elastic constants account of the symmetry associated with an On number of elastic constants degenerates to two. The Equation of Continuity.FUNDAMIONTAL EQUATIONS AND SOLUTIONS the tractions is 5 given by the thnje terms The equations inertia of terms —p motion are obtained by adding all the forces and the for each component: dû/df AxAyAz. • • • .. d% 'df d'w ^^^ + 17 + dp. dp. Poisson's ratio a. w oi a. (1-9). For an incompressible medium.-. In liquids the incompressibility k If very whereas has only moderate values for gases. Ideal Fluid. By neglecting these products. If = 6 = div v = = or. This corresponds to k = fm 0. point in an elastic solid: P ^ -^ = W /> (X + . . v. mean it pressure. „ dw fdu . For many solids. we can write the equations of motion in terms of displacements u.2 pA . the hydrostatic or large. the medium From is (1-11) and (1-12) we find p^^ = pyy p^^ = kd = an ideal fluid. .r p ^1 = (X + m) ^ + MV^y mV^^ + pF + pZ (1-13) P^ = We difference (X + M)f + have replaced d^/c?^^ by d^/df. and we will occasionally use the Poisson relation X = ^t as a simplification. since it follows from (1-10) that the between corresponding expressions involves second powers or products of components which are assumed to be small. is the value of the remaining independent component of the stress tensor. a liquid is . —p. and cr I.6 ELASTIC WAVES IN LAYERED MEDIA fi: Lamp's constants X and - - . where —p. X and fi are nearly equal. we linearize our differential equations. by Eq. dv\ We also could have written these equations using any two of the constants: bility k.. or the coefficient of incompressiThe relations between these elastic constants are given by the equations (tE (1 E 2a) + a){l ^ 2(1 + X a) „ _ k m(3X + fM 2m) _ 7i 1 o\ = X + Using Eqs. \ d9 m) ^ + mV u + . (1-7) and (1-11). dp/dt is = the rigidity ^ vanishes. Young's modulus E. 5-6. (1-13) reduce to PT7 = dt du dd k— dx . writing a^ = k/p. exists: differential a velocity potential defined — - £1^ d!p _ W = Tdx If the y = d^ Tdy _ W = ^ dz dip . Imperfectly Elastic damping ticularly of elastic Media. required. pp. At a free surface of a solid or liquid all stress it will components vanish. see Birch 88-91].FUNDAMENTAL EQUATIONS AND SOLUTIONS incompressible. into the equations /li + m' d/bl in is the equations of motion. Eqs. where 1/Q = wm'/m. k ideal fluid 7 = oo and <t = 0. motion in an may be obtained from (1-13) with = 1-3. dv Pi: dt = dd k— dy . This equivalent to stating that stress e'"' is used. In the problems which follow be assumed that solid elastic media are welded together at the surface of contact." from "internal (For a discussion. dw p-. being without significance. . of k] Reduction to Wave Equations.5. Since the rigidity vanishes in the liquid. (1-1-4) body forces are neglected.In many cases Q may be treated as independent of frequency to a sufficiently good approximation but the more detailed discussion which this case requires is given in Sec. parfriction. m(1 1-4. Boundary Conditions. some special conditions must be added. If the medium to which the equations of motion are applied is bounded. y.. implying continuity of all stress solid-liquid interface slippage can occur. At a and continuity of normal stresses is and displacements alone 1-5. and displacement components across the boundary.) of The effect of internal friction may be introduced motion by replacing an elastic constant such as ^t by is [9. For simple harmonic motion.= dt dd k— dz . and reduced to one as follows. The equations /i of small 0. tangential stresses in the solid must vanish at the interface. we easily see from (1-14) and (1-15) that ^ and where the additive function = = t a^d + f Fit) (1-16) ^ of a" d dt (1-17) Jo is omitted. the time factor internal friction is introduced by replacing and the effect of by the complex rigidity + ilQ).^. a Unear function of both the strain and the time rate of change of strain. ju The equations motion of a fluid [derived from (1-13) with = and therefore X equation if = can be simplified <^. These conditions express the behavior of stresses and displacement at the boundaries. . We shall also be concerned with the waves resulting from imperfections in elasticity. (1-15) . Now. we can write the first of Eqs. separate wave equaof a disturbance (ii = 0) motion.-=1.2. (1-13) in the form dx V ^ dy V df) dz V df It is easy to see that this equation in a similar form will be satisfied if and the two others from (1-13) written the functions (p and are solutions of ^}/i the equations VV = 4^ VV. ^3) (1-20') By the definition of 6 as given by (1-5). d we obtain (1-21) = VV In general. it (p and a vector potential ^{\l/i. in dx dy vector form. with potential For displacements in a solid body. s(w. where tions are obtained for these forces may dfl two types of motion. V. under the assumptions mentioned above. w) and il = i curl s [see Eqs. v. This wave fluid equation holds for small disturbances propagating in an ideal velocity a. = i^ . Assuming that the body be neglected. However.2 8 ELASTIC WAVES IN LAYERED MEDIA the definition of From mean pressure p P ! =--kO dip and (1--16) we have (1-18) V - in Then. is 1A2. by introduction of the potentials <p and xp^.3 (1-22) . the equations of motion (1-13) represent the propagation which involves both equivoluminal (6 ^ 0) and irrotational 6 = div s(m. w) = grad (p + curl t|r(i/'i. \p2. from (1-16) and (1-5) we obtain 1 d'^ (1-19) in which small quantities of higher order have been neglected. dy dz = 1^ ay + ^-^ dz dx (1-20) dz or. (1-2)]. convenient to define a scalar "As) as follows: _ djp d\j/3 _ dip dx . -^ + ^T^r) a? - dTd-r) ^ 'Je /^^ d^q d^w\ d^w " ^\dz dr drV ~ df of The angle x does not appear. dcp d(rW) r (1-26) . Spherical symmetry does not persist during propagation in a layered medium. (1-25) are satisfied of equations VV where . and in the second equation the function W may be replaced by another . = V'W-'-. = 4^ at . + . the two equations of motion are ^ + ^l (X + 2m)(^-.. we now use the following: q = ^ dr d<P . However.W r p y-^ dt (1-27) d d V=7-2 + r-7+ dr dr . „^. Instead the potential defined in (1-20). Love [31] discussed expressions which represent solutions satisfying the equations of motion. x are the cylindrical coordinates (usually taken with the z axis and w passing through the source and normal to the layering) and q are the displacements in the r and the z directions.^dz it dW ^' = + -^-^ ^ dz dr . The first equation in (1-27) is a wave equation. in many cases axial symmetry will remain. and we shall make use of cylindrical general case. respectively.2 . If r. in the most it was pointed out by Pendse [42] that the use of scalar and vector potentials is not always free from ambiguity. I d 7-^ dz in cylindrical coordinates. In most problems to be considered in subsequent chapters. (iij a and /3 may be propagated through an elastic Wave equations involving the dilatation and rotation • • • directly V9 = -2T72 a dt Vfi^ = T2 -772p dt (1-24) can be readily derived from (1-13) or (1-22). spherical waves from a point source will be considered.FUNDAMENTAL EQUATIONS AND SOLUTIONS where a 9 (1-23) = J^—^^ /^ = a/- The wave equations velocities (1-22) indicate that two types (9) of disturbances with solid._. By if substitution from (1-26) <p may the functions and W are solutions a be shown that Eqs. because of the axial symmetry.- ^ - . z. but coordinates. (1-22). Murnaghan [40].e. waves (also known as shear. strains. Eq. By introduction of the potentials (p and yp to that of solving the wave equations. for an arbitrary number of sources with arbitrary positions and for any conditions which can be imposed at the boundaries of the medium. The problem of finite deformations of an elastic body and the effect of high initial stress on wave propagation were discussed in a series of investigations by Hencky [21].. = i^ r. 5. (1-29) upon differentiating with respect to that W satisfies the second equation (1-27). Stokes' theory concerning waves of finite amplitude is discussed in detail by Lamb [30]. As a i its form is somewhat complicated. one function satisfying we have only include additional effects such as those of body forces possible to It is Scholte Chap. Biot [6]. A more direct procedure would start with the general solution of the wave equations and adjust is this solution to the conditions of the given problem. The usual method of obtaining a solution for a given problem has been to superpose certain particular integrals. By making if trial substitution = A exp y {vix + V2y + U3Z =F ct) (1-30) . forming a sum which satisfies all conditions of a given problem. equivoluminal. (1-29). Solutions of the Wave Equation. if ^ is a solution of the wave equation VV we see immediately. 1-6.10 function xj/ ELASTIC WAVES IN LAYERED MEDIA defined by IF = -f (1-28) Then. Plane Waves. the general solution will be considered after the discussion of simpler cases. The wave equations (1-19) and The distortional \{/ (1-22) are linear partial differential equations of the second order. When there is axial symmetry which must be of \l/i. For a fluid we have reduced our problem we obtained a single ance. and distortional waves with velocities a and /3. i. It obvious that a general solution must be an expression which holds for all cases of wave propagation in a homogeneous and isotropic medium. transverse. Birch [7. 8]. or solutions Eqs. wave equation representing the propagation of a compressional disturbTwo wave equations were found for a solid. representing the propagation of compressional respectively. Stoneley and (see finite or 4) applied the classical theory of elasticity in papers concerning gravity waves in compressible media. Several additional examples are given in Chap. rotational) are represented in general by three functions. and Keller [27]. V2. similarly for C. and we must obtain two different values of velocity. I From t condition (1-31) we see that v-i may be considered as the that. If direction of we take the z C and D. Ai = A2 = 0.A3V3) + M^3 = The which B is A. where in the direction of L. while C and D have mutually perpendicular direccoefficients may tions. for a given time direction cosines of a line L. Then (1-33) take the form obtain the conditions I'l = j^s = 0. Thus Eq. ip is constant over any plane (normal to L) determined by the equation v^x length I. advancing along L with velocity c. if c = a and (1-31) + pI +4 ^ vi. we For the component B we put in (1-33) A3 = B. L with wave periodic in time with period l/c. axis in the direction of L. . C. in order to simplify Eqs. V2y v^z + + T is <p sinusoidal is aionji. will determine components of the displacement vector s. A^ = A3 = 0. It is instructive to substitute as trial expressions for the displacements in the equations of motion (1-13) (body forces being omitted) the values of w. Equation (1-30) shows = tn.Vi + + + + . = C. be considered as the resultant of the vectors B. fJi)v3(A. -pc'C -pc'D -pc'B + + + ^C = îD (X = (1-330 2fi)B + = 2ju)/p is associated Thus we see that the velocity c = a = the component of displacemen t pa rallel V (X + with in to the direction of propagation and the velocity c = /3 = Vm/p is associated with the components any two mutually perpendicular directions normal to it.421^2 + -]- A3V3) A^Pz) + -\- mAi iJiA2 =0 = (1-33) A2V2 A2V2 4. A2 = D. A^ = A3 = 0.v.v. do — P and Furthermore. and for D. at any given point. D. namely. w given by the equations u V = Ai exp exp • • i —j- {viX + V2y + v-^^z T • ct) (1-32) = A2 • w = A3 exp • • obtaining the three homogeneous linear equations in A^ -pc'Ai -pc'A2 -pC'Aa + + + (X {X (X + -{- tJ^)viiA. (1-30) represents a system of plane waves of wave length I. A. When we make a similar trial substitution in case of a solid. the direction of advance depending upon the sign chosen. pi)v2{A. the x and y axis in the respectively. 1^3 = 1. v.FUNDAMENTAL EQUATIONS AND SOLUTIONS 11 we easily see that the wave equation (1-19) »'? is satisfied. a and /3. (1-33). there are two wave equations (1-22) to be satisfied. . (1-33) directly for c" Had we in c^ with a single root at a and a double root at by elimination of A. SV (vertically polarized shear). is. origin. t).^ [Qi.iR - at) + f2(R + at) (1-37) and /z are arbitrary functions. Thus the system of plane waves traveling along L consists of three independent parts which correspond to the P (compressional). A similar treatment of the remaining equations in (1-22) yields 'A. The For spherical waves in a fluid we would start with Eq.- = ^ g. Spherical Waves. Thus. (1-35) we obtain the equation d't} 1 d'^ dR^ a df (1-36) and it is evident that its general solution has the form ^ where /i = f. takes the form (p 2 dtp 1 5V dR R dR = ^^«' '^ (1-34) By the substitution . a cubic /3" would have resulted.(B.iR - m+ g. If we write the x^ -\- first equation of (1-22) in spherical coordinates.12 ELASTIC WAVES IN LAYERED MEDLA solved Eqs.. and SH (horizontally polarized shear) waves of seismology.2 correspond to disturbances traveling in the opposite direction and are usually zero. solution of the first equation of (1-22) f = ^ UîR - at) + f2iR + at)] (1-38) Each term in (1-38) has a constant value on a sphere R = const at t = ôIf /i and /a are periodic functions. (1-19) and obtain for the velocity potential ^ where /i = ^ [UR - at) + UR + at)] (1-39) and J2 are again arbitrary functions. (1-38) will represent infinite trains of spherical waves propagating toward and away from the common center of the spheres R = const. by (1-35) and (1-37) the in this case. putting R^ it = d y^ -\- ^ and assuming that ip = <p{R.2{R + m] (1-380 The functions /j and represent radiations from a point source at the functions /z and ^. Sommerfeld. Terms Oil) in (1-38) are excluded by this "condition of radiation. k being a parameter To prove formula solution of the (1-41) Sommerfeld pression for the right-hand a representation of exp made use of a more general exmember." such as fiiR extended to three dimensions a similar condition derived Bakaliajev [1] propagation of elastic waves in two dimensions. a further condition is necessary which specifies that no may be radiated from infinity into the region of sources. The factor in (1-40) depending only on distance was given by Sommerfeld [55 or 57] in the form Jo(fcr)6-'''' R where Jo z and r v' Jo f k dk ^^ V (1-41) = = = zero-order Bessel function cylindrical coordinates k' — kl. given. by Kupradze for + Special solutions of the type (1-39) satisfying the radiation condition are the functions <p = 1^ exp [±:i{kM - cof)] (1-40) where A = ka const o)/a — A detailed discussion of the mathematical expressions representing is radiation from a source. together with proofs of uniqueness. and Weyl. Since each product Jo{kr)e~'''^^ is a if wave equation ( / = k^ — k^. we can attempt to derive to — ikaR)/R by superposing such expressions obtain r F(k)Jo(kr)e-'"'' dk In order to determine the coefficient F(k) we can consider the values of both functions at z = 0.FUNDAMENTAL E:QUATI0N. Thus we put F{k)Joikr) dk Jo f (1-42) . a condition usually imposed that the disturbance vanish at infinity. and is it The radiation condition guarantees was shown by Haug [20] that this holds even if [59]. ka Very important transformations of expressions representing spherical waves were used in the theory of wave propagation by Lamb. in the work of Stratton a unique solution. According to Sommerfeld energy [56]. for example.S AND SOLUTIONS 13 is When sources of a disturbance act in a finite domain. complex in expressions such as (1-40). and these will be applied in the next chapters. +.y + v...3. calculating the last integral. v^ = cos ??. and integration along the imaginary axis gives ^(*> = -h L we ^ find da + k cos a Now. while ?? is to be taken complex and to vary from to then 7r/2 and subsequently from 7r/2 to 7r/2 + z'oo. V2 If we put sin sin cr. < cr < 27r. d'L = sin M§da.v\ -\. we apply the Cauchy theorem to transform the path of integration from the real axis to the infinite arc and where the latter integral is obtained for the Bessel function imaginary axis of the first or fourth quadrant.z)] dX (1-45) fExpression (1-41) can be derived from (1-44). according as the coefficient of ^V in the exponent is positive or negative. To perform the integration with respect to r. 14 ELASTIC WAVES IN LAYERED MEDIA left-hand The member can be written in the form of a Fourier-Bessel integral = '^ / k dk Jo{kr) j e~'^"'"Vo(^T) dr Jo Jo Hence F(k) = k f Jo k_ "^ l-K I \ e-"'''Joikr) d Y J-^ da \ if 1 I — ka+k cos ir(-k \T( e ff) \ J di Jo by substituting the expression (1-69) and reversing the order of integration. .x 4.v.xH-. Instead that the of using cylindrical coordinates.(. Weyl [64] considered the expression (1-40) as a result of superposition of plane waves and proved first exponential factor in (1-40) can be written in the form of a double integral 1 r -a-.f As an extension of Weyl's integral. In this expression the integration must be performed over the half sphere 1^3 > of the unit sphere v\ -\. thus proving the relationship (1-41)..) -ikM ^2 (1-44) Ztt J where rfS is a surface element. The contribution from the infinite arc vanishes. the expression j'l ^^""^^ ^^ = -^ / F'[at - {y.vl = 1 = = & sin ?? cos a.. by Duvall and Atchison [17].FUNDAMENTAL EQUATIONS AND SOLUTIONS 15 which represents an arbitrary spherical wave. The relationship between these functions and those of Stokes and Rayleigh was also shown. These formulas represent the displacements at a point produced by a given distribution of initial values. was given by Poritzky [44]. Recently Ilzhevkin [47] suggested the use of new functions to make the study of the energy transport in a field more convenient. If a pressure p(0 applied to the surface of a spherical cavity of radius Ro in an elastic medium. who considered displacements in a field due to a pulse at the boundary of a spherical cavity for a = I. Solutions in the form (1-40) are used for an infinite medium or for a certain time interval in a finite domain until the effect of boundaries has to be considered. The potential of this field was expressed in terms of a set of special functions. and Selberg is [51]. Another transformation of the integral representing a spherical wave propagating from the wall of a spherical cavity (R = Ro) was given by Selberg [51].aj T Ro dco t-\- dr (1-46) which for t a generalization of (1-40). {\ = fi). in good agreement with several observed characwaves near explosive sources. F is defined as a i((al function of its real argument but F' must also range over complex valu(!s of its argument. [26]. Certain applications to elastic waves produced by a source similar to an explosion were studied by nates were recently derived by Kawasumi and Yosiyama [54]. t) = is 1 2TrR LL -77l( A(co)p(T)êxp R ?. and by Blake [11] for general where = teristics of values of Poisson's constant (X ^ m)- . Sharpe obtained the approximate formula for radial displacements " ^ Rlpo sin R — 8[ t Rq for t 2V2 ixR M > R < R — Ro (1-47) a = 8 for Ro t 2'\/2a/3Ro. Sharpe Blake [11]. Other applications of solutions representing spherical waves were made by Barnes and Anderson [2] to explain the phenomenon of the so-called "tail" in pulse propagation from a spherical source. The first representation of th(! sound field produced by a spherical source of harmonic oscillations was given by Stokes and developed by Rayleigh [45]. Sezawa [52]. the resultant wave can be represented by the equation ip{R. For p(t) = po ior t > and p{t) = < 0. In this formula. Sezawa and Kanai [53]. Some general formulas for the displacements in spherical polar coordi- Homma [22] for cases important in seismology. differentiation in the direction of the outer normal. The average value M„{<p) of a function on this sphere is given by the equation al cos -&. y. y + a^ sin ?? sin r. • • •) sin t? dd^ dr (1-48) Poisson's solution of the <p{x. at <p (1-50) F is interpreted [5].) (1-49) It may be proved by substitution that this expression satisfies the wave t equation. A solution was given by Poisson. where § and r are two spherical coordinates. Hadamard [19]) a sphere S having its center at the point P{x. the closed boundary of D.16 ELASTIC WAVES IN LAYERED MEDIA Poisson and Kirchhoff Solutions. These are <p{x. for example.t) = a \^. tion A point of this sphere has coordinates x z -\- -\r al Bin d^ cos r. Then Kirchhoff's formula. It gives the value of the function ^ at a point in terms of initial values of this function P at a moment and its derivatives with respect to time at the distance a = al from P. Let us now denote by R the distance PQ. A function <p(x. for P inside D. z. y. z. the value of the integral zero . The classic problem of sound propagawas given the following mathematical form by Cauchy: Values of a function (p{x. (See. y.{^. is Cauchy problem was obtained by considered as a mathematical form of the Huygens' The Kirchhoff formula can be written even for a more general than case that considered above. (p and its first derivatives are continuous and that the second derivatives and F are finite and integrable. z) and a variable radius a = al. y.) as the source-density distribution. Consider the inhomogeneous wave equation VV + where Fix. by S. I) wave equation may be written in the form = J. for example. 0) = <po(x. in this region. l) is sought which satisfies the wave equation (1-19) and the initial conditions given above. Another form of the solution of the Kirchhoff and principle. [lMX<po)] + IM. y. y. z. y. t) and its derivative with respect to the time are given at an initial instant t = 0. z) and [d(p/dt]tô = <pi{x. Let us take (see. Q being any point in the region. z). any given point P in assume that. Bateman terms of In order to obtain the value of at its values in a certain region D we have to [F] indicates t that the value of the function F is to be calculated at time — R/a. z. y. and by d/dn.z. M„ = Then — // (p{x + at sin ?? cos T. is dip -For = ///f--/f^'l(l)-^ dn is J_dR aR dn d(p Tt dS (1-51) P outside D. We see that ^{t . If we assume that *S is Kirchhoff's formula may A general discussion of conditions which hold at the wave fronts as well as an extension of the Kirchhoff formula and of solutions representing the propagation of a disturbance in an infinite elastic medium has been given by Love [-32.FUNDAMENTAL f:QIJATIONS AND SOLUTIONS 17 For the homogoncous wave equation.v) dudv (1-52) Another form of the solution of the wave equation was given by Bateis itself an arbitrary function of number of parameters can be easily increased if desired. Whittaker's form [66] of the general solution is <P = I fixsinu cos I V -\- y s'mus'mv +z cos u -\- at. y.i) is a solution of (1-19). The be con- . x. Let [3]. provided that 2 v\ -\.vl -\. z. man Since the integrand in (1-52) coordinates and time and a solution of (1-19). Previously we made use of the plane-wave solution (1-30). t for the argument of the function /. We can now take the most general linear expression in X. This corresponds to Huygens' principle in that Q may be regarded as a secondary source sending disturbances to P. ' An arbitrary function (1-54) . (1-57) will fWe assume sidered later. A general solution of a partial it is may be written in different forms.V3. are eight parameters. d/dn = d/dR. z. and sometimes impossible to transform one into another.vl = -^ a (1-55) This condition may be written either in the form v^ = ±\/-2 — = ±a \/v\ vi — vl V2 (1-56) or 1/4 -{- -\- vl = ±1/ case of complex v.{t - t) (1-53) where Vi. be reduced to Poisson's formula (1-49). a sphere of radius R — at with its center at P. depends on the value of v? and its derivatives at points Q on the surface at a time preceding the instant t by 11/ a. y. the X = v. General Solution of differential equation Wave Equation. For a periodic function of time. VuV2.{x t - x) + PoXy - y) + ' • vXz - z) + v. and dcp/dt = -a d<p/dR. u. which is the time for a disturbance to travel from Q to P with the speed a. the volume integral representing the so-called retarded potential vanishes. a simpler formula of Helmholtz is easily obtained. here that these parameters are real. 33]. f fix. For this case we note that ipf.R/a) = ^(0). V2. Vi. /2(X2.{x - x) -\- V2{y - y) -{- a/-| - a vl - vl{z - z) + + v^{t - i) (1-61) X2 = "1(3: x) -\- V2{y y) . Vi. z. z = h. they contain all particular cases considered in the literature. z. vo. t) dvi di>2 dvi dx dy dz dt (1-60) /a are again two arbitrary functions but = v. y. v^. instead of (1-52). V?. we have t to 0. V2. X. cp conditions of a particular problem.{x 1/2(2/ + + Viiz vîz - z) - - i) (1-58) — z) + v{t - i) and <P write. Taken together. general solution. If the source begins to emit a disturbance at = we also put = and write.-. by (1-63) and (1-61). given by Jardetzky [23]. y. y. X. i) dvi dv2 dvi dx dy dz dt + where Xi /i // and /2(x2. Xi Then we can make use 1^2(2/ of two arguments v{t = vii^ - i) x) + -\- - ^) y) X2 = v. z.18 ELASTIC WAVES IN LAYERED MEDIA first Let US consider (1-57). in this solution By to a specialization of functions involved all we can adjust them used. t) dvi dv2 dvs dx dy dz dt + /•••/ where fold. v„ -• .\\A - v\ - vl{z z) Vi{t - i) We can also consider special functions /. vi. vx.--. If a source located at a point S(0. h). . = / • • • / /i(xi .t)dv. are general solutions of the wave equation. and we obtain ^ — I ' I fiixi'. put X = y = ^' 0. X.fV (1-62) we obtain Fourier integrals (p = I • • I Pe"^ dvi dv2 dvi dx dy dz dt (1-63) Both expressions (1-59) and (1-60). Some special forms We is shall show first how the well-known expression for the potential for a single source may be derived from the 0. For example.. /. writing one term only / = P{v„ ••• .dt (1-59) and /a are two arbitrary functions and both is integrals are seven- The ' ' condition (1-56) used in many investigations. r. Jo f P.) j e'" "=°= ''-"' dr [61]). we have to put p = TtV. p.. Including the factor l/27r in the arbitrary function P. possible to write <P = [ J-co PoXp. = ^^1(^)^2 (^4). Pi(k) = - (1-73) V . j.^-v ( l-oo) r sin P2 = < t < tt Then ^ (1-64) takes the form = f e-"' dv. in order to obtain the expression of the form —^ = where R^ j e Jo{Kr) —^ (1-/2) = r^ -\- (z — h)' corresponding to spherical waves emitted by a point source.. (1-68) The last integral represents the Bessel function (see 1 -*- Watson (ff+e) 7 J/ o(Kr) = \ - r"' / '^r cos <r \ e da 7 = — 1 -*- r" / iKr co3 f e da (1-^9) /-i /^r\\ where e is arbitrary. J e''''-"'P{K. We can then write dK = \ e'"*' dp. in we put cos X y = = r u u p. J's = ±\/~4 — K^ (1-67) is independent of r. = K k cos T sin r 0<«:<Do — tt /. dp.i'.FUNDAMENTAL EQUATIONS AND SOLUTIONS ^ 19 - \ \ \ I'(i'^.) P(k.. p.)e'-"' dp. It is apparent. medium. that the function <p P is also independent of because of the assumed isotropy of the r. dp. ri-64) Even though the disturbance has spherical symmetry. we obtain ^= If f e""' dp.)e"—— '-"'*-" having dp. Thus.) dK dr (1-66) By (1-56) and (1-65). p. [ [ e''-'-''^''' "'"' '''"'P(k. we consider now view subsequent only symmetry with respect to the applications. z axis.) J o{Kr)e'' ''''-'' it is dK (1-70) we assume that P(k. f v.{K)JM)e""''~'" dK (1-71) Now we (1-41) are able to see that. " Cambridge University : Press. pp. Trans. H. Schairer. pp. Appl. 1952. 11. J. 522-530. Soc. Moscow. Bakaliajev. Phys. Bateman.. Sphere and the Barnes. Trans. 12. vol. of Mathematical Physics. K. 24. M. vol.= r^^Mg-''-^^/'')' Jo tC is v^. 251-458. : The Variation of Seismic Velocities within a Simplified Earth Model. Am. 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Sommerfeld. 59. 53. A. 5. Math. A. H. P. 862-867. pp. (London): Kept. 28. Sîsm. Ergeb.S.). Apvl. B." Mainka. in Phys. 1948. D. Stoker. 42. 39. 311-321. Bull. 1943. F. pp. 51. 1950. Am." in Heins. 1952. 1945.S. vol.. 37b. Soc. Soc. Berlin." McGraw-Hill Book Company." p. Inst. J. Rayleigh. Schermann. 1952. Sci.: On the Energy Motion in the Field of Spherical Sources. the . Naturw.. 10-17. 45. 52. Mechanics. Macelwane.. 1942. vol. Sezawa. [7].S.: "Some Remarks on Radiation Conditions. 1954. 5. 43. (S. I. : The Propagation of Elastic Waves by Explosive Pressures. 1951. 97-108. Theory.. 1933. Math. pp. pp. 97. 41. 49." in B. J. Tech.: "Vibration and Sound." in "Symposium on the Theory of Electromagnetic Waves. New York. Sezawa. 50. H.: "Electromagnetic Waves.. Newton. Interscience Inc. pp. 118.R. X.\nalysis of a Small Arbitrary Disturbance in a Mag. New York. R. 477-493. 19. 1935. (Tokyo). pp.: "Evidence on the Interior (ed. A. ELASTIC WAVES IN LAYERED MEDIA Macelwane. J. Bull. Schelkunoff. 40. Rzhevkin. 58. VI: Seismology.: Mathematical Questions in Seismology. 14. 55. J." D. C. Murnaghan. Poritzky.. Earthquake Research Inst." 2d ed. S. . pp. 66. pp. McGraw-Hill Book Company.. "On the Wave Equation and the Equation of Euler-Poisson. vol." in Heins. vol. Vanck. 1950. New York.: Sur le probleme de Cauchy pour I'equation de Pois. Soc. Inc. A. 97-119. 1952... 1954." 4th ed. Watson. iiber einen Inc. pp. R." vol. B. 64. Whitham. J.: tion and Vibration Theory. 481-500.: 23 "Electromagnetic Theory. A. N. C. . G. Watson: "A Course Cambridge University Press.: The Propagation of Spherical Blasts. 1952.FUNDAMENTAL EQUATIONS AND SOLUTIONS 59. 1941. vol.: "A Treatise on the Theory of Bessel Functions. G. E. "Proceedings of Symposia in Applied Mathematics: Wave Men- 63.. 1953. London.: A Contribution to the Theory of Elastic Waves Produced by Shock. 203. Acad. 571-581.son et I'equation des ondes. 234. (Lomlon) A. Cambridge University Press. N. New Czechoslov. (ed. 3. 65. 61. London. and G. of Modern Analysis. H. 5. vol.. E. Ann. Phystk. 62. T." 2d ed. Phys. 60. 1919. Paris.). pp. Set. 1952. 60. Proc. A. Whittaker. J. Weinstein." McGraw-Hill Hook Company. Weyl. Stratton.: Ausbreitung elektromagnetischer Wellen ebenen Leiter. Roy. 2584-2585. York. Weinstein. J. A. 2-1. these will affect the form of solution. in a direction or P e with the boundary with the normal to it (see Fig. the plane P and SV waves do not depend i ^0 in the xz plane which makes an angle e = 90° — on y. Eqs. The case of incident waves is represented in Fig. and we shall begin with the simplest problem of this kind. 2-2. Various types of homogeneous and isotropic half space will be discussed in this chapter. 2-1 and that of SV waves in Fig. 2-1. We assume that the free boundary of a homogeneous and isotropic medium is a plane {z = 0) and that a train of plane waves propagates waves generated in a Fig. new must be taken into account. Reflection of P wave at free surface of elastic solid. For the reference axes chosen. At the instant the disturbance reaches a boundary. The displacements corresponding to these waves (2-1) dx dz dz dx 24 . We shall consider two types of plane incident waves. (1-20) form two separated groups.CHAPTER 2 HOMOGENEOUS AND ISOTROPIC HALF SPACE The solutions of wave equations considered in the preceding chapter all represent disturbances propagating in media of infinite extent in conditions three dimensions. 2-1). Reflection of Plane Waves at a Free Surface. 2-2. when solutions of (2-3) having the form <p = f(z)e' yp = k(ct-x) g{z)e (2-4) are used. to satisfy the boundary conditions at 2 = 0. both potentials ip and \p must be used and. the exponential terms must be identical. It will be shown later that.HOMOGENEOUS AND ISOTROPIC HALF SPACE will be considered together. in (2-3). for example. the >S7/ component V = dz dx (2-2j representing a pure distortion can be treated separately. (p The functions and r/' satisfy the wave equations (1-22) (2-3J Fig. we obtain (2-5) and the integral of this equation is f(z) = Ai exp iik-J-^ — I zj -{- A2 exp ( —ik\l^ — 1 z (2-6) From these results we can write the solutions (2—i) in the convenient . Reflection of <SF wave at free surface of elastic solid. Substituting (p. Considering the case where only in Eqs. (2-7) and (2-8) that if B2 = 0. + 2. we obtain for distortional waves 1 = tan f and c = a sec e = /3 sec f. P Waves. 1 — X (2-7) = Bi exp ifc( cf + ^\^ — + I — xj ^2 exp ikict — z ^-1-x To interpret these expressions we note along the surface.. Thus. 27r Let k = 2-K cose I cos f r where I and V are wave lengths for compressional respectively.I2 dx~ dz' = (2-8) [p„L. and distortional and distortional waves. AO = that c is an apparent velocity a represents the distance traveled front KA in unit time.f XVV + 2.26 ELASTIC WAVES IN LAYERED MEDIA form (p = Ai exp ikict + 2-v/^ — \ — xj + rp A2 exp ik\ ct ^— q. waves with emergence waves with emergence and reflection angle Incident incident. Then (2-7) represent compressional e and reflection angle [.|? dx + «Î dz = = . Incidentally. In Fig. and A2 = —A^. in general. which is similar to Snell's law. = or e = 7r/2. = = it . 2-1. and it follows that by the compressional wave — 1 = tan e In a similar way. OK = c is the corresponding distance traced by the wave front along the free surface. an wave produces both ~° incident compressional reflected compressional and shear waves..(§ + £i^ exists only if e may be seen from Eqs. . (2-7) we set B^ = the remaining coefficients by use of P waves are and determine the relations between the boundary conditions dx dz b. . <p X. and the boundary conditions take the form 2(Ai (Ai - ^2) tan e + S.776 or 2. cos' = 3 cos' f. for reflected these two special directions of an incident wave no P wave These roots correspond to c//3 = 1.000. (2-1). for grazing incidence. In order to measure the angle of emergence e from seismograms. The angle e = tan"^ A„/Ah is called the apparent angle of emergence. e (2-9) If cr = J. .HOMOfiENEOUS AND It follows LSOTUOJ'IC JIALF e 2 SPACE given above that 27 from the definitions cos' of tan and tan cos' f f e = J. and. (2-9) and (2-10) the ratios A2/A1. one makes use of the amplitudes A„ and A^ of the vertical and horizontal ground displacements. 12°47' or = 30°. and (2-11) we obtain t™ Walker [54] ' = 1+3 tan' 2 tan f e tan' = ^^ f — 1 = -'"' e . see that B2 vanishes for 7r/2. for normal incidence. In both cases the reflection consists of a f P 2 wave only. higher order with respect to tan e the denominator being of a f and second. By Eqs. the coefficient A will vanish e)' 4 tan (3 tan' e e + = 2)^ = (1 + e 3 tan' This equation has two roots exists. _^ + A2)(l + 3 tan' + 2B. if Since tan' = 3 tan' e e + 2. B2/A1. tan = f From Eqs. = 0.ftan' e) f - 1) = [ . 2f (2-12) in the . (2-7). therefore. and e: can be expressed in terms of A2 Ai B2 Ai _ ~ 4 tan 4 tan e e tan tan tan f f - (1 (1 + + = + + + 3 tan' 3 tan' e) e)' e)' (2 11) _ ~ -4 4 tan e e (1 f + (1 3 tan' tan 3 tan' e)' From the second of Eqs. two cases. First. (2-11) we e.^ . ^. e = vr/2. then a^ = 3/3'. derived the relation between 2 e and sin for a' 9^ 3/3' form 2 cos' e = ^ (1 - e) (2-13) which he calls Wiechert's relation and which for Poisson's relation a = j takes the form 2 cos' e = 3(1 - sin e) (2-14) This equation also follows from (2-12) and (2-9).__. X = M. 788. (2-7).28 If ELASTIC WAVES IN LAYERED MEDIA a 9^ 3(3^.000. = [ 1._ . 2-2). partition in the system of incident and reflected waves. Incident S Waves. we write the corresponding particle velocities in the form U — d"(p/dx dt. Taking the kinetic energy per unit volume as |p(w^ + w'). Again assuming that Poisson's relation holds. for incident and reflected u u = = BJc^c tan f cos kxi f w = w = B-Jz^c cos kxi —Bzk^c tan x. km/sec. Making /3 use of the value a/^ = 1. we may compute the energy flux for the waves mentioned above by multiplying . (2-7). One can easily derive the expression for the Bo does not vanish but has a small minimum near apparent angle of emergence for the case of incident SV when o- = |: For an incident is reflected as SH wave a similar derivation shows that all the energy SH. It Let us now consider an incident SV wave (Fig. Similarly. (2-15) 3 tan' 3 tan' e)^ e)' The amplitude and If f of the reflected distortional wave B2 vanishes 1. w = d"<p/dz dt for P waves. = 58°. Partition of Reflected Energy. and substitute the other terms in the boundary In (2-7) we put Ai = conditions (2-8).01 e.776 for f = 55°44' f = a/0 60°. Then /cxi ^^ u u for incident = A^h^c cos = — A^'c tan e e cos ^'xi = Aok\ cos kxi w = Aok'c tan cos kx2 SV and reflected waves we can write P waves. _ and x'i are the expressions in parentheses in Eqs. the conditions for vanishing of A 2 are changed. cos kx2 Bzk^c cos fcx2 In these equations. corresponding to a = 7. _. Eqs. = 4. we obtain the foUomng equations for the reflection coefficients: A2 Bi _ _ ~ 4 tan 4 tan 4 tan 4 tan e [ (1 f + (1 (1 3 tan' e) tan + - B2 Bi e e tan tan f f + = (1 + + + 3 tan' e)^ . also corresponding to c/l3 and 2. The boundary conditions are satisfied if the incident transverse wave gives rise to a reflected transverse wave and a reflected longitudinal wave.788. the horizontal displacement of the free surface being To derive an expression for the energy twice that of the incident wave. Walker found that A 2 will not vanish for any real value of attains a small minimum near e = 20°.17 km/sec. . among others. a. b. 7 2 B2 Bi tan f an incident SV wave. Eqs. I = a' + b' (2-19) where for a^ = A' ^ Ai and Bl tan 6' f A f tan c an incident P wave a 2 or - A. the corresponding ecjuation e ^pBlki^c^^ sec^ f sin f = \pAlk*câ sec' sin e + If the hpBlkV^ sec' f sin f (2-18) energy flux of the incident waves is taken as unity in both cases. and the expression \/(^ la the surface. Reflection at Critical Angles. It follows from Eqs. — 1 corresponding to the potential becomes negative imaginary. (2-17) and (2-18) reduce to the form. whereas Gutenberg prefers the square roots of the energy ratios. Thus we may write the equality between incident P-wave energy and reflected P. B2/A1. e c and . exists until f Since a cos f = /3 cos e. etc. Computations for reflections from a free surface have been given by Jeffreys [15] and Gutenberg [10]. indicating that for an incident SV wave there is no reflecte d P wave i n the range < f < cos~^ i3 a. useful for computations.HOMOGENEOUS AND ISOTROPIC HALF SPACE the total energy per unit volume (double the 29 mean kinetic-energy density) and the area of the wave front involved.and *SF-wave energy for unit area of the free surface as by the velocity of propagation \pAlkVa sec^ e sin e = ^pAlk^câ sec^ e sin e + For the case of incident ipB2'/cV/3 sec' f sin f (2-17) is SV waves. tan f = \/c'//3' — 1. and a > c > ^. Jeffreys presents his results in terms of the reflection coefficients A2/A1.. 2-3 and 2-4 Gutenberg's calculations for various assumed values of the ratio for a/l3 are reproduced. The motion (p decreases exponentially with depth from . tan e = —i's/l — c^/a^.2 tan p2 . Within this range. In Figs. (2-15) that the amplitude ^2 becomes complex with B2 = —Bê'^'^ where tan 8 ^2 _ c'/^r (-"-0^ This represents total reflection with no change in ampHtude and a phase change in the reflected SV wave The coefficient Ao also becomes complex. no real value of e reaches the critical angle cos"^ (3/ a. and the expressions (2-7) fail to represent e trains of waves. To see the reason and to . find another form of the potential. the general integral of (2-5) f(z) = Ce^' + C^ze'' = Ci + C^z (2-22) as well as for (see Jardetzky [14]). the function f{z) must satisfy the ordinary differential equation (2-5). (2-4) equations (2-3) in and the form (2-5) If we write the solutions of the wave (2-4). and the roots is ti = T2 ~ 0. Now = the roots of the characteristic equation of (2-5) are Tl. = (c = a). we go back to Eqs. ^2 = -Bj = 0.30 ELASTIC WAVES IN LAYERED MEDIA Fig. This holds for an incident P wave an . = — Ai. Square root of ratio of reflected to incident energy for P wave incident at free surface for various values of Poisson's constant.2 = c zLik 1 ) ztik tan e (2-21) For grazing incidence. {After Gutenberg. c = 0. 2-3. Under these conditions.) where A2 A critical case occurs for the grazing incidence of P waves. = a. Square root of ratio of reflected to incident energy for free surface.' HOMOGENEOUS AND ISOTUOI'K. (After Gutenberg. Waves satisfying the second condition are called surface waves.„ - 1 ' j to 90'^ / \\ •ill' PtoO 1 f/ // // ' \\ \ \ \ ^// Q V"' 1 i// if i'/ \^ to syto 1 \ \ \ \ \ f i \ 1 / / incident e / 10' \ \ \ \ yj > 1 \ 1 wy 't?\ 20' 30' \ to 90° 1 40' Fig. Free Rayleigh Waves. A solution cor- . 2-4. Rayleigh [39] gave the theory for surface waves on the free surface of a semi-infinite elastic solid. f From the equation tan^ f = 3 tan^ e + 2 it follows for = that tan = ± \/2.) &V wave incident at SV wave.. and the expressions (2-7) can be written in the form (2-23) derived by Goodier and Bishop [9]. 2-2. showing that the motion became negligible at a distance of a few wavelengths from the free surface. IfALF SI'ACE --'V^. A solution of this form represents a wave with amplitudes increasing with distance from the interface.— 31 ^^ '^ \\ V> ' y yT yr ' 1 ' i sv. Assume a simple harmonic wave train travehng in the x direction such that (1) the disturbance and (2) it is independent of the y coordinate decreases rapidly with distance z from the free surface. ^ „. Eqs. For an incompressible a —^ <» ^ and (2-29) reduces to | + 2l|. Thus for either upper or lower signs '-Wh c' *(' . it we use and Eqs.1 B = (2-27) J|-14 + (2-|)b must = In order to have values A and B different from zero. the SH component is .?)tl ff (2-28) The quantity c^/(3^ can be factored out after rationalization. p. . and the Rayleigh equation takes the form o c .. Since we assume that the plane boundary of a half space is a free surface. Hence this solution is not of interest. 2/24 16 . c Now the second factor in (2-29) negative f or = 0. always a root of Eq. (2-24). B. V = C exp ik{ct ± a/^ \ z — X (2-26) The arbitrary constants A. the stresses must vanish at ^ = 0. From the conditions p. In order to find the expressions for the stresses. and under these conditions surface waves can exist. p.' 32 ELASTIC WAVES IN LAYERED MEDIA responding to this definition may be obtained from Eqs. ^ 5)] If c = (2-29) = 0. = 0. (2-24). (2-25).-16 = (2-30) . (2-25).y = 0. (2-29) solid if < < and is positive for c = /S. (2-24) and (2-25) are independent of time. /S < c a. 2 follows that C = - |. (2-8). and (1-11). and from (2-27) A = —iB There is and u ^ w = is 0. and (2-26) can be determined from the boundary conditions. C in Eqs.)a ± 2^1 . (2-7): I z (p = A = B exp ik{ ct d — — — — X f or (2-24) < 2 < 00 \p exp c ik\ ct ± ^' \ z X (2-25) provided that given by < z approach zero as ^ < a and the sign is chosen so that the potentials approaches oo In a similar way.^ = 0. the parameter c satisfy an equation obtained by putting the determinant of (2-27) equal to zero. (2-26).S < a. The last root corresponds to the velocity Cn 2-2/ = 0. a = 3/3.9194/3 (2-31) The other roots of Eq. Simon. The displacements decrease continuously below 40 ft. (2-30'). (2-28) and satisfy (2-28) except for a change in sign. The last root alone can satisfy condition 2 for surface waves. Despite the departure from homogeneity'. The other two roots for this to surface waves with velocity c case are complex and do not represent surface waves. These roots arise from squaring Eq.136 of a wavelength.2 + 2/ V3. V3.bme-''''''') sin k{ct - x) it w = Z)(-0. The determinant corresponding to this new equation is obtained from the boundary conditions if we consider and A^ = B2 = 0. Values of c//3 The second wave and a computed for these special cases in Sec. . since the radicals in (2-24) and (2-25) become positive imaginary. It may be seen from (2-32) that the particle motion for Rayleigh waves is elliptical retrograde in contrast to the eUiptical direct orbit for surface waves on water. For the condition (2-31) one finds for the displacements u = D{e-°''''" - O. (2-30') correspond to real values of the radicals and (2-35) and therefore do not represent surface waves as mentioned above.HOMOGENEOUS AND ISOTROPIC HALF SPACE 33 This cubic equation in c^ has a real root at c = 0. solutions given by (2-7) for the two cases A2 = Bi = in (2-24) The first case represents a compressional wave incident upon the free surface at an angle such that only reflected shear waves occur.8475e-''''"'' + 1.91275/3''. The crossover depth was 0. 2-1 from the general expressions for reflection coefficients are identical to the values given by the extraneous roots of Eq.192 of a wavelength and reverses sign below this. They found that the motion is retrograde above 40 ft and direct below. and (2-29) becomes = V c" c 56 c 32 This equation yields three real roots. corresponding 0. see Fu [7].4679e-°'''''0 cos (c^ - x) where Z) is a function of k. The vertical displacement is about one and one-half times the horizontal at the surface. case represents the reverse situation of an incident shear reflected compressional wave. Dobrin. and Lawrence [3] experimentally determined the particletrajectory variation with depth below the earth's surface for Rayleigh waves from small explosions. Horizontal motion vanishes at a depth of 0. the experimental results offer good agreement with the theory. For a more detailed discussion. These experiments were conducted in a region of layered unconsohdated and semiconsolidated rocks. If Poisson's relation \ = n holds. cV/S" = 4.95/3. the more important points of it will now be discussed in some detail. (2-29) for all possible values of Poisson's ratio. 0. 2-3.c '' -D QD \ \ 0.r> / \ O Kj. n o / "^ / -0 13 Qd.2 0.2 0.. Cu/a for Rayleigh waves from Eq.) Calculations on wave propagation in a half space using plane-wave concepts as illustrated above are inadequate for problems in which characteristics of the wave source must be considered. and solutions in Lamb's paper form the basis of much of the material in later chapters. Lamb He considered the disturbance generated in a semi-infinite medium by an impulsive later studied force applied along a line or at a point on the surface. Ratios Cr/^ as functions of Poisson's ratio. Since the methods . [22] first In a classic paper.34 ELASTIC WAVES IN LAYERED MEDIA Knopoff [20] has computed the ratios CrI^. These are shown in Fig. CrJo.4 Fig. In this section we consider a two-dimensional problem and derive expressions for surface displacements arising from a force which were and others. Lapwood [25]. also wrote the formal solutions for internal sources as integrals by Nakano [28]. ^ a /-> \ U. Surface Source.4 /S/a. 2-5.4 A A \ n o. Integral Solutions for a Line Source. 2-5. {After Knopoff. using (2-8). where the functions and yp are solutions of the wave equations (2-3). the signs in these equations correspond to the lower signs in Eqs. and in order to find their values we now make use of potentialsf where the amplitude e'"' is ^ which satisfy the = Ae-''-'"' lA = Be-''-'"' (2-34) wave equations kl (2-3). and 5. The stresses p.AkW (2-38) Rayleigh's function. Henceforth the time factor omitted to save space. are given by Eqs.HOMOGENEOUS AND ISOTROPIC HALF SPACE 3o The displacements u and w (f applied normal to the free surface along a line coincident with the y axis. b..=o = b.o = Ze''"'-"' Z depends only on k.^ and p. (2-1). (2-24). we obtain _ where is 2k' - ]^ Z{k) j. are given by (2-8). + fx)l of Lamb [22] is used._2ikv Z{k) F{k) ^ (2^ - klf . We now wish to superpose an infinite number is of stress distributions of the form (2-33) such that the resultant a concentrated line source. we put Z from — 00 to + 00 = —Q ..=o = Kx) = -£ f" = e-'"' dk (2-39) Then if we put in (2-39) /: some f{k)e''' d^ -Q(k) [i(pt (2-40) tBecause exp [{(ut — kx)] instead of the factor exp of our expressions differ from his in sign. is The effect of a periodic force applied perpendicular to the free surface expressed by the conditions (2-33.J.kDA + — k^)^ 2Bikv' = If the conventions v' v = (A. = ^ p (2-35) A and B are functions parameter k as specified by conditions (2-33). On inserting (2-34) in (2-33). Solving these equations for A and (2-27)..J. (2-25). we obtain 2Aivk {2k' - {2k^ - kl)B = Z(k) ^^ (2"^^) . To do this. dk/2ir in (2-33) and integrate with respect to k obtaining b. provided that v" ^k' - v" of the ê - kl k^='^ a fc. .^ = ik{c / a — 1)' and Re i* > ^yith similar ones for are attended to.]. In order to obtain a concentrated normal force at a: = 0. = 0. force j{x) along the x axis vanishes everywhere except at x assume that the normal = 0..hY is of second kind of zero order This particular form chosen because. Then Eq. This may be proved by a method not unlike that used for the . diverging outward with velocity function may and Eqs.36 ELASTIC WA\^S IN LAYERED MEDIA the Fourier integral we obtain /(•x) = ^ ZTT J _ oo C ^""'^ ^^ f J - Z®^''' ^^ CO (2-41) representing the distribution of the normal surficial stresses.kj- 2vv') _a-x . as iQ r'" k{2kr . will be understood. (2-1). of this Lapwood has given a more detailed discussion problem. where \kaR\ is large. (see Watson [56. r°° = iTe""H^'\kM) = -2e'"' Jo e-"'"-^' coskx— ^ rlh (2-44) transformation Hereafter the factor e'"\ although not written. (2-41) reduces to (2-39). A solution of the wave equation representing compressional waves propagating cylindrically outward from a line source at a. with Q a constant. using Eqs. where it that /I" /(^) d^ = — Q is finite. the displacement of any point in approaches infinity in such a way the surface 2 = may be written. (2-43) represent cylindrical waves a. With the stress specified by (2-39). Methods of evaluating integrals such as (2-42) will be discussed later. (2-34). ^°=-2^Lf(^^ ^^ The effect of a concentrated horizontal force P acting parallel to the X axis at the origin can be represented in a similar way. as well as a generalization for an impulsive source. s = h may be written as ^0 = iire'^-'m'îk^R) ^Po = (2-43) where Hq'^ R' = Hankel function = x' + (2 . Another transformation of the Hankel be used to write (2-43) in the form 198]). Internal Source. and (2-37). p. Lamb [22] also [25] considered an internal line source of compressional waves. we j' can make use of the functions <Por = —4: Jo e ^ cos hx dk iô = (2-45) formed by adding to (2-43) the potential source (0. 37 three-dimensional transformation (1-41). we obtain for a compressional source <p = ^Q \ "F7TT e""^''"^''^ cos kx dk (2-47) _g r K2/c^ . ^^. ^.^. (2-44) in order to ensure the vanishing of ô as z -^ oo Now. In order to insure that i/' satisfies this condition. Then. at the free surface. (Por \l/ we require Re v' > Now. Hy a simple -2 r COS kx e-" ^ Jo V = -f e-'-'"' — J-co V (2-44') Comparing (2-44) and (2-44') with (2-34)... In order to make the tangential stress 00 from our choice Re 0. D.. 2._. —h) corresponding to a reflection at the free surface.. solving for A and D and substituting in (2-46). substitution of the functions + ^ and in the boundary conditions (2-8) leads to two integrals which must vanish for all values of x. Jo F{k) .^ ^.. we can obtain an interpretation of the former in terms of the superposition of plane waves. HOMOGENEOUS AND ISOTROPIC HALF SPACE corresponding transformation. Since the coefficients of cos kx and sin kx in these integrals must vanish separately. in order to have vanishing normal stress p. T^^ ^_. of an equal and opposite image found that p. If the it is functions (2-45) are inserted into (2-8). we add to (2-45) the potentials and \f/ in the form -\- (p = 4 Jo (A cos kx B sin kx)e "' dk (2-46) ^ = 4: (C coskx Jo -\- D sin kx)e''''dk The necessity for superposing functions such as is 9? and 1/' in order to obtain vanishing stress at the free surface of the incident connected with the curvature wave front corresponding to ô.C. One finds first that S = C = 0. We note that the convention Re > for v given by (2-35) must be adhered to in Eq. we obtain four equations for A. .All can be satisfied by incident and reflected P and SV boundary conditions waves alone only when the waves are The first integrand as 2 plane.B.. as in Eqs. We have free see from Eqs. : 38 ELASTIC WAVES IN LAYERED MEDIA for Equations (2-47) together with (2-45) complete the steady-state solution an internal line source of compressional waves. Symmetry about the z axis (taken through the source) is usually assumed.(kr)T.(z) (2-53) provided that the functions S and T satisfy the equations ^ + -^ + dr r dr (^^ + ^^)^ = o (2-54) . The case of a point source located at the surface of a medium has more applications than the two-dimensional problems considered in previous sections. V^ = |i dr and particular solutions ^ of (2-50) i|r dr + (2-52) may be taken in the form = S{kr)T(z) rp = S. These values of k are also singularities of the integrands in (2-47) A method of evaluating integrals of the type (2-47) will be given in Sec. Surface Source. \p.. Integral Solutions for a Point Source. (2-37) that for Z = is. parallel are represented in terms of potentials ^ and and perpendicular to the z axis. and we can put semi-infinite r = Vx' + y' u = -^q v = ^^q (2-48) The displacements w and g. 2-5. 2-4. Let us assume that the primary disturbance varies as a simple harmonic function of time. the coefficients A and B finite values only if F{k) = 0. For an internal source of shear waves the corresponding functions may be found in a similar manner. dz dz^ f df ^^ ^ and we again assume a time variation the wave equations (1-27) (1-29) take the reduced form (V' + /cf)^ = = {V' -h kl)rp = (2-50) fc„^ ^ a + kl = i p |^ dz (2-51) For the axial symmetry. that for those values of k for can which Rayleigh waves are possible. (1-26) and (1-28) ~^ ^ If ~ dr ^ drdz ^ dz dr" r dr e'"'. (2-54) has a particular solution is satisfied T = e~'^. (2-57) we get for the stresses in the plane 2 = 0.Uo = = m[2^"M m[(2/c' - k{2e kl)A - kl)B]J. To form the resultant displacements we note that the variable x corresponds to r cos 6 and integrate (2-34) with respect to 6 from to tt. Similar equations hold for (S..HOMOGENEOUS AND ISOTROPIC HALF SPACE with v"^ 39 v'^ = k"^ — k't. Equations (2-56) and hence (2-58) may be also derived from Eqs. k is real and < k^ oo < k. with = k^ — k^. whereas the sum of displacement components in the r direction must be taken. Thus we have i-ikAe^'' + - S/e""'^) [ e . v and v' must be positive The solutions (2-55) vanish as 2 ^^ and also vani. by multiplying bj^ cos 6 before the integration is performed.{kr) = - dJn(kr) d(kr) The stress equations in cylindrical coordinates are ! /dq P.. 1\. the first is a form of Bessel's equation which by the Bessel function JJkr).ikr) (2-56) w = — vAe""' + using the relation k^Be'"'Uoikr) J. [p^rUo \p. with \ = n. = ) \e + dw 2m (2-57) dz where d = VV = . real.sh as —> By 00 because of the property of the Bessel function Jo(kr).. The displacements in the z direction are obviously directly additive.ikr) ^2-58) - 2kVB]Joikr) by Eqs. (2-1) and (2-34) for a line source if we consider the effect of an infinite number of line sources in a uniform arrangement about the z axis.d r dr ( dip\ d'ip V drJ ^ dz' With Eqs. Thus we have two (P particular solutions = Ae""Joikr) If x^ = Be"''\/„(kr) /c„ (2-55) A r and B being two constants.r = dw" P. dividing the integral by the length of the interval w. (2-49) we have q = —(kAe ( "' — kv'Be-' 'iJ. The second of Ecjs.dd cos 6 "" — (2-560 w = {-vAe~'' dd Bike-''') Jo [ IT . (2-56). the equation F{k) = 0.2kVB = - where F(k) = (2fc' - klf . The function F{k) is identical to (2-38). Writing the time factor e'"' again. 2-3. Appropriate boundary conditions for this case are Ip. this result becomes identical with (2-56). (2-37) was replaced by ikB. Z = we can put (2k' A = where - kl)C B = 2vrC = 2\/k' . on writing k = k. being an even function of ^ and Ji(^) an odd function To obtain the solution for a point source we now consider a force per unit area ZJo(kr) acting normal to the free surface. In this case the constants k of zero..^kW may be compared to (2-37). Z = from is 0._(^^. since The expression (2-61) B given by Eqs.. {z we have by (2-56) the surface displacements = corresponding to free annular surface 9o waves = -Ck(2k' - kl - 2vKVR)Ji(Kr)e"' (2-64) Wo = CkjvjiJoMe"' . as be seen from the expression for a Bessel function of the nth order may JJ^) <ô(^) = — TT [ Jo e'^ ''°' " cos nd dd of ^.=o = ZJoikr) (2-59) Then by (2-57) and (2-58) we have -2vA (2fc' + (2fc' - k'.J.)B = (2-60) - kl)A .klC 0) (2-63) C is a new arbitrary constant. Free surface waves correspond to and B in (2-55) have values different as was mentioned at the end of Sec.-) .40 ELASTIC WAVES IN LAYEEED MEDIA If B is replaced by Bik. Z ^ ^2-62) J (1 Z_ P{k) n FREE ANNULAR RAYLEiGH WAVES. Then. The displacements form (2-56) at the surface z = can now be written in the ^ _ M2e _ k^v -14F(k) \ 2.rUo = b. if the parameter A. A equal to a root in (2-61). b..ô = -^ / — J'^(kr)k dk (2-67) To obtain the solutions for a surface point source. I fia)2ir(x da = -L (2-66) Then (2-65) reduces to [p. by (2-65) origin. fc'( 2/c' kl - 2VV') ^_.{kr)k dk f J fia)J.. (2-67) which. p. 559) to represent the stress distribution p^. (2-68) in the form ^0 = ~7r~2~ 27r fx A Li I r" cosh U . for FORCED WAVES.) We may now u rewrite Eqs. to 0° "° ^ J^ 2tm f k\2k' r Kizk" kl kff — 2vv') 2vv ) T n \ ji ^ i„ Fik) ^2-68) *= By verified that [p^r]z=o -2^/. u du Ir" J_ . Jo ^^''^ (2-70) r" f" klkp -ikr coah .J.2 41 = = k 2 — kl 7^2 Vr K —Kb In ordcr to pass to a concentrated force Le'"' acting of the Fourier-Bessel integral (see. . at the origin.(ka)a da f2-6oj a. Ref. we can make use example.: HOMOGENEOUS AND ISOTROPIC HALE SPACE where vl . we can put Z = — Lk c?/c/2x in (2-59) and (2-62) and integrate with respect to k from . (2-68) may be = and [p„]^=o reduces to Eq.b u ^j.J. 18.. I) Now where suppose that /(a) vanishes for all but infinitesimal values of it becomes infinite in such a way that Jo is finite. for example. and (2-66). eo. 56.A. satisfies the condition for a concentrated force at the In order to compare these results with previous ones for two dimensions we use order: the relations which hold for Bessel functions of the zero and first J^{kr) = -"^ ''' f (e"" ""''' " - e-""- ^°^' ") du (2-69) J^(kr) = -TT Jo p.fc. Ref. 1(1 •'»»'•)* it finding the surface stresses corresponding to Eqs. f {e'" ""^^ " + e""" '=°^' ") cosh u du (See.=o = J{r) = f Jo J. 180. let R = vr^ -{.J-0 [p. considering an unlimited solid in addition to expressions (2-55) for z > we now need the following < 0: <p = AVJoikr) rp = BY'^Joikr) (2-72) The conditions for stresses on the plane 2 [p.r]+o = are (2-73) - [p. (2-8).^ be the distance from the source and write the potentials ^ xP = = '—^ /i = e-' / Jo(fcr)e-''-" ^^ V (2-71) Jo The transformation potentials in which r (1-41) enables us to use the integral form of the and z occur in separate factors.2kV(B + kl){B 5') = = ^ on (2-74) -2v{A -\- A') -f (2e - B') Similarly. On 0. = r cosh u Q = L In order to obtain the potentials corresponding to an internal source in a half space. and (2-72) we obtain = m {2e - kl){A - A') .-^' = -4 ^ - «' = -2^7 (2-76) .{kr) = for X By Eqs. This problem involves a normal periodic force 2 ZJo(kr) per unit area acting at the plane potentials for ^ = 0.42 ELASTIC WAVES IN LAYERED MEDIA see that the definite integrals with respect to k in Eqs. In practical applications this solution represents the effect of an explosive source in a medium.AB + 50 = A') - k'(B - B') = From (2-74) and (2-75) it follows that ^ . provided that the wave lengths considered are large compared with the diameter of the source and small compared with the distance to the nearest boundary. CONCENTRATED FORCE.r]-o = ZJ:. To represent a point source of compressional waves. COMPRESSIONAL-WAVE SOURCE IN AN UNLIMITED SOLID. the condition of continuity of displacements 2 = leads to (2-75) AviA -{- A'. (2-70) are We obtained from the integrals (2-42) for the two-dimensional case by performing the operation —d/wdx and substituting X Internal Source. we again follow the method of Lamb. (2-55).J+o b. (2-72) and (2-76) we obtain for 2 < (2-79) ^ = -—^^ 47rco f e'"Ju{kr)k dk 47rw p Jo jhr e'''J. .ikr)^ V p Jo l)e (2-80) The expression (1-41) may -ikaR It used to transform the potentials <f and \p in Eqs. e-'''.L. 47raj 2/ p Jo e~"'"Jo{kr) ^-j'' by using Eqs.(kr) we (2-77) To represent a concentrated force Le'"' acting at the origin to follow the procedure of the previous section. one originating at a source »S(0. A suitable expression for the be obtained by extending the solution of the previous section to the case of two equal and opposite forces constituting a couple. —h).e-"Jo(kr) 4^ = ^1^. As might be expected. (2-55) now take the form 0) = ^. (2-81). Thus from (2-71) 0 (2-81) which represent the solution of the problem as given by Stokes [52].— HOMOGENEOUS AND ISOTROPIC HALF SPACE and Eqs. COMPRESSIONAL-WAVE SOURCE IN A HALF SPACE. 0. Put and integrate with respect to k from Z = —Lk dk/2ir in Eqs. h). Of particular interest in the study of earthquakes is a source which simulates the failure in shear that is known to be the principal action at the foci of earthquakes. we obtain (p = - —— I e '"Jo(kr)k dk for 2 4x0) p Jo > (2-78) = Similarly. the other apparently originating at the image point *S'(0. this potential may be derived b}^ taking suitable potential may derivatives of Eqs. Let us consider a potential (p representing two spherical compressional waves. (2-77) oo Using k^ = aj//3 and m = (3^p. . problem of finding the disturbances Next we consider the produced in a half space b}^ an internal point source of compressional waves.. 0 r ^''^%1^ ^'^ e-"/o(fcr) dk 2-5. . which are part of the solutions (2-70) for a surface point source.rUo = [?). A useful approach is to replace by the complex variable f and to use contour We shall evaluate integrals of the form . (2-59). which occur in the solutions (2-42) representing the surface displacements Our results may be also adapted to the integrals with respect to k. are found to be e-''J.kl .. By lo = = 4e'"' j -2e'"' YnX'(^ "''Jxikr) dk (2-86) .44 ELASTIC WAVES IN LAYERED MEDIA For < s < /i we put {z — h\ = h — z and obtain <p = 2 Jo cosh vz e-'^Joikr) ^-^ ^ x{/ = (2-83) From (2-49) and (2-83) the displacements at Qo 2 = w. by and evaluation by numerical integration is exceedingly the variable of integration k integration in the f plane. Evaluation of Integral Solutions. (2-55). by performing the operation —d/wdx and substituting X = r cosh u. difficult. (2-61).]. The integrals obtained as solutions of the problems treated in this chapter cannot be evaluated direct integration. To obtain potentials which will annul the residual stress of (2-85) we put Z = -2^.{kr) Jo k^ dk '^-^ V = (2-84) The stresses at the free surface determined by the potentials (2-83) are (2fc [v.V -ê-'kdk in Eqs.kl) e-''^ rfr r(2f .=o = 2m f Jo V M is g-Vo(fcr)/c dk (2-85) Thus an additional system of surface stresses required to satisfy the condition p^^ = at ^ = 0. Application of Contour Integration.kl s/f.2Vf kl Vf -kl klf -4rVf (2f kWf kl ^ (2-87) for a surface line source. from and (2-62) and integrate with respect to k adding the displacements (2-62) with the changes mentioned and (2-84) we obtain the resultant surface displacements to 00 . the intek of grand *(f) or ^(0 ) has real poles determined by the zeros F{^). The of the radicals v Re J/' > in and v' will be determined by the conditions Re agreement with the signs used previously. . — For a complex w = s — zV = 0. first the complex values of w also have to be considered. To simplify the further discussion we shall assume that there is only one pair of branch points i/c^. 2-7. restricts the choice of a cut. For a real co. the conditions (2-89) take the form kr = k^ - r' <^. according to (2-91). Thus. The second condition defines the part of the hyperbola to be used as a cut. k^ = (s — — r" + 2zA-r — (s" — and ia)/0.se choices infinite values of ^ as 2 —^ 00 are avoided. However. these cuts being parts of hyperbolas.signs f"^ = ±V — /c^. This is accomplished by introducing cuts in the complex plane. The imaginary axis determined by the conditions (2-92) is not an independent cut since it does not pass through a branch point.since with i/ > the. that is. Re i^ = -sa/a k^ - r^ < (s' - a^)/a (2-89) The cuts first of these conditions shows that for s > the branch points and must lie as in Fig. Re > . (zbA>. it can be used if it is combined with that part of the real axis to form the cuts AOE and BOL. as there are four combinations of signs of and v' . v (±/o„.Jie a" v' = we have k^ ^ kr {s requires that /c^ — ia)/a. 2-6.HOMOGENEOUS AND ISOTROPIC PIALF SPACE 45 We note first that when f is a complex variable. The Riemann v surface for integrands in (2-87) and (2-88) has four sheets. = /c ± it. Therefore we will take the cuts for complex k^ = w/a and k^ = w/^. the denoniinator of the integrand defined by (2-88). f' are introduced by the radicals = ± Branch points a/^ ^ — A:^ and existence of branch points requires that the integrands be made uniform functions before Cauchy's theorem is applied. selected according to the requirements The permissible sheet must be Re > and Re v' > 0. 0). f (zb/c. (2-90) They show that either 7 = = z^ ^' < ^5 (2-91) or k -t' < a ^ (2-92) Thus. 0). (2-7)]. j' the cuts will be given by Re v = Q. since the condition use. a part of the real axis we can between the branch points — k„(B) and k^{A) [see Fig. The . where v^ = ^' — kl. or 0. 2isa)/a be real and negative. a = 0. In (2-87) and (2-88) the branch points ±/c„ and ±/c^ are located on the real axis but when the methods of operational analysis are applied. as shown in Fig. For CO real. on Ak in Fig. right half plane. from the transformation r = fc„ + 5e'' (-TT < < tt) (2-93) where 6 is measured from the k axis. for any permissible path in the right half plane. The cuts AOE and BOL appear then as hmiting cases of those parts of a hyperbola (Fig. Im Thus. 2-7. Re v does not change sign on any permissible path in the i/ = 0.46 ELASTIC WAVES IN LAYERED MEDIA Im v = J^^^^ Im 1^' = Fig. for ^ = h > k^. in may be neglected. . 2-6) which are cuts for complex w. that is. 2-6. Only Now. + V2^ e'"' for a small region = \/2k^ 5 which b^ (cos I + I sin |j (2-94) around A. Im v can change sign only on crossing Ak. Branch points and cuts in the complex ^ = k -\- ir plane for Re co > 0. We shall now evaluate the surface-wave terms. and principal values of integrals is must be used. negative on the right side of the cut in Fig. the singular points of the integrand are on the path of integration. Following the same procedure as for real co. Im v discontinuous along the branch being positive on the left side and To determine the signs x measured from A T. while Im j/ > in the first and Im i' < in the fourth quadrant. the tangent to the hyperbolic cut (see Fig. For complex is Re i' = v is on the first part of a hyperbola determined . will displace as was done by Lamb.HOMOGENEOUS AND ISOTROPIC HALF SPACE 47 Lkr Re c>0 Im i'>0 / Re ./ = 4 / (m 1/ = 1 Re Im v>0 V <0 E Fig. A and B and cuts AOE and 50L in the complex f plane for From the foregoing we see that for all permissible paths in. Residues. Lamb [22] demonstrated that Rayleigh waves are the largest disturbance at a surface point far removed from a surface impulse. The assumption that A-^ co complex the points k^ and as well as the roots k of the equation F{^) Therefore. of Im V. (2-89) The Im zero in the second part given by (2-89) line. introduce the angle — tt < < If we consider the integrals (2-42) for a real co. by Eqs. 2-7. 2-6. seems to be more useful to consider first the contour shown . 2-6). it = from the real k axis to a line whose slope is Im co/Re co. we may determine the signs as shown. he found other terms representing waves which diminish more rapidly with increasing distance from the source. Branch points real w. In addition. the right half plane Re i/ > co. . 48 in Fig. 2-8. we can easily - kl - 2vv') /: = 2iTiHe-''' F{k) + f $(f) rfr + + [ <S>(r) d^ (2-96) kjv -ikx /: where Fik)' dk = 2iriKe-''^ + f ^(f) d^ ( ^(f) rff (2-97) H = k{2k' - kl) . Now taking. Integration path in the complex f plane. exp axis { — rx) makes the integrals zero.2Vk' F\k) k! Vk' - kl (2-98) K = . + + { = 27^^• ERes exp ( the infinite arcs because of our choice of the contour in the lower half plane. we see that f On H^)d^ =+/+ NH and GM. + [ + f (2-95) There is only one pole (k) La and L^ are branch line integrals. for example. the presence of the factor exp — ilx) = +00 — ikx) • -00 1 k \ V \ \ \ \ \ \ \ \ \i \ \ N N " w X " "! III -' ^"•>- ---' w VG ---i-jjlu/t---- H Fig. Therefore the integral along the real is [ $(fc) dk = -2x^ X)Res in this case. M{x)/N{x) at a pole x = find that by Eqs. ELASTIC WAVES IN LAYERED MEDIA 2-8 and then to take the Umiting case w real.klwK F'{k) . the integral (2-87). (2-87) and (2-88) /c(2/c' and the integrals along the loops Using for the residue of an integrand a the expression M{a)/N'{a). The velocity is that of Rayleigh waves. we first examine for real co the last two terms in Eqs. the ratio of vertical to horizontal amplitudes being K/H. This ratio is the same as that obtained for free waves in Sec. 2-9. It is seen that the orbital motion of a surface particle is retrograde elliptical. (2-99) result from the contribution of the pole in the integrands (2-42) and represent a train of Rayleigh waves traveling away from the source with an amplitude of displacement independent of X. To obtain further information about the waves represented by the branch line integrals. The first line integrals have been omitted for terms in Eqs. This loop is shown in Fig. with branch points fc„ and kff on the real axis. Branch Line Integrals: Line Source. we obtain Uq QH = —t exp [i(o}t — Kx)] -\- (2-99) Wo QK — exp [l(a}t — KX}] -\- where terms derived from the branch the present. 2-9. £ formed by contraction of cC^. and £^ for w real. 2-2. produced by a surface line source. . (2-96) and (2-97). In this case. since [see /c is a root of Rayleigh's equation F(k) = (2-38)]. the two loops L„ and L^ degenerate into one (L). The cut AiAOH obtained ka H G Loop Fig.HOMOGENEOUS AND ISOTROPIC HALF SPACE 49 Applying these results to the integrals for the displacements (2-42. for example.. Eq.f .50 ELASTIC WAVES IN LAYERED MEDIA is by contracting two hyperbolas characterized by the condition that its Re v' = along the entire cut and Re i' = along part AOH. v'... For the case oj real we contract these contours but maintain the separation between the loops and adhere to the convention of values for p and v' on the different banks of these loops as given in Fig. p.ZLM^LA'V^l-xfc (2fc^ . 2-9). line integrals of the type (2-87) are reduced to the term (2-100) ^ ^ ^^ _ - f'^ j^^ v v'. 8kl [ k„ (2k — kff) — e-'"' IQkvVf dk (2-103) This expression was given by Lamb [22. 2-10. H^v ' "^^ we see that the integrand is a function of the product w'. The more rapid the factor exp { — ikx) for increasing x imply diminishing (icot) velocities ranging values of the integral (2-102). v' > < in first in fourth quadrant quadrant (2-101) Since v'. Writing now./ > 0. The integrals of the type (2-88) cannot be v reduced similarly because the factor stands alone and the parts from is HO A and /.. vp' On the part of the loop HOA the product has the same value as on GCA (Fig.^K F(-^V) I ' • itfl 2kl 2k. and the corresponding parts Thus. If we put z^ = i*.fc^)^ - 2vv') . (2-100). the branch of the integral along L cancel each other.ikW. for example.f \(2F . = (k' — kly is real positive = Im = Im = — J'' dj.hlf . Now under the conditions given earlier in this section the imaginary parts of v and v' are discontinuous along the corresponding cut. GCA 2kl no longer cancel. In order to obtain expressions which yield more information about the aggregate of waves (2-102). 2-8). The resultant value of (2-88) = d(-zr) + -'di-ir) f '^S^. 17]. we return to the two loops L„ and L^ (Fig. _ _2fc!. near OA^ Eq. for (2-87) ^{2f -kl/.4fc>/. (2-100) can be written in the form The product of exp waves travehng with fluctuations of and (2-102) can be interpreted as an aggregate of from a to /3. Im v and Im v' are positive in the first quadrant and on the left side of the cut OH and are negative in the fourth quadrant. (2f . v'. must be replaced by .]C. = Im . where L^11_ZlA^J^L.l r j^ê-'^dk ^ 2/ + K 7 2 V V. 14 ./.r.k.V.^'vy 2vv'. la kl)' - 2pp: 4/cV. .2vy .kl . 2-10. Afu. 2k' .^ }4f + 4f^.kl + 2vy _ .kl)' ..H^y / 2k' îo (2F 1(2 2/c' .2vv'.kl . i>e-'''kdk 4/c'»/. >0 <0 V above i'' = lm >0 I'' below < above below above below AAi AjCO Fig. .kl)' .^. 2k' {2k' (2-105) integral (2-105) can also be written in the form L-fJ "^ if 2f - /c| (2r kl)' + + 2vy H'^/r 2k' {2k' {2f 2f .ktr .kl . ...klf + 2r' (2-104) _ . y-y^kdk {2k' . <0 V (loop £J v' [''(loop £^) HO or HC resp... Specification of signs of for f = Re = Re >0 >0 when loops v'=Im / i/' >0 <0 i' V v' = Re 1^' >0 contracted Im v and Im £„ and =C^ are w real.kiy + 4/b'. \W which is I vv. 2uy.2vy k dk . . v and v' is attended to. _ .i'. On the upper bank on the side of the lower bank of L„.Izir - 2vy...J4y - 2vy.>0'eft <0 ./.kl)' - e-'''kdk (2-1050 4A. . 4/c v. 2k' Jo kl — 2vv'.HOMOGENEOUS AND ISOTROPIC HALF SPACE 51 L'L + and i: (2/c^ 2f^ (2r^ . + The - /c| S2k' L.v\ 2^ {2f 2h' (2A. we have Im i'>0 Re v= Im />0 !m C-P- Re i^'=0 v=0 Im v' = ^ -T Im v' A.J the convention on radicals of the loop Lp.^k%y 2k' . CA .kl + .kl-2 ik-'-^d^ .kiT .kl .AkW. = Im f' right >Oleft < right OA or resp.-.0 J 2r^ /c| + 2vy {2f r^° . = lm.' - ^L±_2^î__l.v. 27. G(0) = 0.-^p + . and v = v' v = ^q v from A.. In order to obtain their approximate values. p. In order to show this. it takes the form 4Jcle'""-' f w'G(w)e'"' du (2-107) where /^r._..G(0) r(5/2) G'iO) r(7/2) G"(0) + -^^ ^j+ -^. fFormula (2-109) can be derived as follows: First we observe that the major contribution to this integral is due. yields x~^^^. in the case of the integral (2-106). (2-109)t Now. in general. dr (2-106) the second integral in (2-104) substitute u = k^ — k. Now.„ to /c^ A:^. 12(A. . r9-ins^ It can be easily seen that at a large distance x the integral (2-106) small because of the factor r exp ( — tx) becomes and that the integral (2-107) becomes small because of very rapid fluctuations of the exponential factor. according to an integral definition of the r(cr) gamma function in Ref. it can be written form t(2t' -4ki f Jo If in + we " kl)vy. Figure 2-10 shows also that on the lower bank of L^ from to the second radical v' has the value = —v'.^) — u) v. (2/ + kir - 16rV>.- = x" / e-"T'-i dr Jo and the formula (2-109) is proved.52 p ELASTIC WAVES IN LAYERED MEDIA V.dr = ri 1^(^/2) „. we can make use of the formula ^~r( G(r)e ^. . to smaller values of the variable r. in the since f = — tV in the first integral in (2-104). therefore./^ _ . Substituting.v.„ — u) — %J — 1d(/c„ . to /c^. /: -^37r. = Im J' < from v'. . however.-i [2(fc„ - n)^ - kl]uyxk. 75. . The integral (2-107). and the contribution of this integral diminishes as x~^^^. (?(r) = G{0) + rG'iO) + 1-2 G"iO) + " dr we can write 00 CO oo j TÎ^G{T)e-'- dr = (?(0) r Jo e-"T(3/2)-i dr + G'{0) f Jo g-^r^^/a)-! CD / Jo G"(0) _| e-rx^(7/2)-l d^ 1-2 i ^ Now... = Im < 0. since for X 9^ the integrand decreases rapidly for rx ^ ] because of the exponential factor. = — = V. we note that for large x the predominant contribution to the integral (2-107) occurs in the vicinity of a term in the lower limit because of the rapid fluctuations of the last factor. we find for the second expression in Eqs. (2-99) takes the form Uo = —- — jj. ^*P-^f^ exp {-i |) (2-U2) In a similar manner. and ^^^^ - {2ki - kir Thus we obtain J La f = Cik^xy'^'e-''"'' -^ 0(x-''') (2-111) where C = -2^2". Upon inclusion of the branch line integrals the first expression in Eqs.. = [(/c„ . d. the approximate value of the second branch line integral (2-105') is [ J Lff = Dik^x^'e-'''^ and k^ + 0(x-''') (2-113) where D depends on /c„ but not on the distance x. exp [i{cjot — Kx)] + - — !C(A. (2-88) and do not depend on the distance x. Then we can apply formula (2-109).7i)* ^2"^^^^^ for u « kc. Wq = —i - — exp fjL [iioot — Kx)] — - — {Ci{k^x)~^ exp [iiuit — • • k^x)] Zirfx + D.HOMOGENEOUS AND ISOTROPIC HALF SPACE 53 We can therefore extend the upper hmit to <» with neghgible error and distort the path of int(!gration to the imaginary axis. noting that V.„x)"^ exp [i(oot — kax)] Zir/jL + (2-99) D{kpxy"' exp [i{p:t - k^x)]\ + • • • (2-114) Following a similar procedure.{k^xy' exp [i(cot - A>t)]} + • (2-115) where C^ and Di are determined for the function ^ in Eq.uf - kl]"" = (-2/c„M + u'f ^ (-2/c. The expressions D. and Dj are written for reference: D = -2iV2ir -^1 - |l exp (-t^) A=-4^V^(l-f exp(-/^) . We shall derive.3535 for X 1.. important terms.. however. in (2-114) and (2-115) represent compressional waves propagating with velocity a = w/lia.54 ELASTIC WAVES IN LAYERED MEDIA According to the formula (2-109). The surface vibrations corresponding to these waves are rectilinear. Lamb [22] gives for the ratio of vertical t o horizon tal {}z\ amplitudes of the compressional waves 0. k^r are large. p. .. (2-115) and integrating with respect to the variable u.^. the function [i{cx)t H[^\Kr) = — 2 r°° / T exp (—i/cr cosh m) cosh w dw = —[Ho~\Kr)]' Jo .^. (2-114) — (iLH/Tii) K exp — kx)] in which. Performing the operations just indicated upon the first term in Eq. the first term of Wq. Amplitudes are proportional to (/l-„. we find LK KB i^t irfx r \ -i-cr cosh u e au -J Jo Using the definition of the Hankel function TT Jo we obtain 2 — Ho (Kr)e Applying the asymptotic expansion for Hq^^ (see Ref... and the third terms represent shear waves propagating with velocity /8 = w/A. As we have seen in Sec.. .. kKL „ (2-117) leads to A similar transformation of the first term in Eq.t)~^ and {k^xy^\ respectively. We can therefore use Eqs. (2-96) and (2-97) to obtain the corresponding expressions (2-70) for a surface point source. 198) Hîz) = xl\ exp ' +k+ we can write the first term in the form .For = M- shear waves the ratio is 2 Vl - = 2 V273 = SURFACE POINT SOURCE. Now to obtain the more we also make use of Eqs. for example. the factors represented by the definite integrals with respect to k in the point-source problem are obtained from the corresponding two-dimensional equations by performing the operation —d/irdz and substituting x = r cosh u. 2-4.^ . whereas in an unHmited sohd the The second terms amplitudes decrease as x"".633 for X — 2ka)/2k^ k^/kj vkl — kl or = m. vd^^K"^-'^^-i)_ . 56. the next terms decrease as x'^'"^. where /c„r and (2-114) and (2-115) with the constant Q replaced by L. r\~^. (A-13) shows that such an integral can be approximated by an expression with a factor \x\'\ that is. (A-1) x = k^r or k„r. \k^r\~^ or \kc. This may be easily seen by putting in Eq. The remaining terms again represent shear and compressional waves with amplitudes diminishing at least as 1 exp ( — ik^r cosh u) du (k^ry exp { — JkaT cosh u) (hry /: 7r(cosh w)" /: 7r(cosh uY du respectively. At large distances these reduce to {k^rY' and (kaJ')''. (2-119) and (2-120) again represent Rayleigh elliptical vibrations with the same ratio of horizontal to vertical axes as in the solutions for a line source. In this case. i\oit KT + + r exp \i{(jit — — kj.HOMOGENEOUS AND ISOTROPIC HALF SPACE is 55 involved. where the factors M. and the Rayleigh waves predominate. namelj'. N. however. where n = | for q^ and n = f for Wq. The functions of k^r and k^T represented by second factors in these expressions are integrals of the type considered in Appendix A. Ci. C.cosh v)] c^w (cosh v)^ exp ik.ry [2"(aj^ /c^r cosh u)\ /: du (cosh w)^ + (2-120) L. which can be combined with the first factors. (A>r)~' and {k^rY^. line integrals from Eqs. the familiar law for divergence for annular waves. (2-114j and we finally obtain for the displacements go = IkLH 2-KKr exp i\ wt — KT — — — kgj- + + and 1^0 M A^ r exp [ijoit cosh^)] du (A. (2-112) hold only for large k^r and k^r. n. Mi. and D^ by Eqs. Then Eq. These expressions The first waves having retrograde terms of Eqs. exp l\(x)l r2-i]8j 2/x Transforming terms due to branch (2-115). Ni can be readily expressed in terms of and (2-116).„r)* 'af) J() Jo (cosh w)^ [ijoit r" exp k^r cosh^)] du (cosh m)' -\r (2-119) = kKL exp w. The amplitude decrease of r~' is also . and it yields the term kLH 2fJL H[ {kt) exp {i(xit) ~ — I kLII . the amplitudes diminish with distance as (nr)'^. D. The path of the second integral is distorted into the contour. INTERNAL LINE SOURCE. 2-5 for co complex. which includes the positive imaginary axis and the infinite arc. shown in the first quadrant of Fig. I kG{k)e''' dk - ^. Following Lapwood. 2-11. 2-11. we note first that the integrals (2-47) have the form h = \ [ G{k)e''^ dk + \ f^ G{k)e-"' dk (2-121) U = where G(k) ^. containing the negative imaginary axis. the loops L„ and L^ around the cuts. and a . Following the procedure given in Sec. we distort the path of the first integrals in Eqs. Another method for evaluation of these integrals was used by Sakai [41] and recently discussed by Honda and Nakamura [12]. [ kGOcje'-"'^ dk (2-122) is an even-valued function of k. shown in the fourth quadrant. (2-121) and (2-122) into the contour Fig. Integration path in the complex f plane for internal line source.56 ELASTIC WAVES IN LAYERED MEDIA more rapid than is the case for elastic waves diverging from a point in an unhmited medium. A detailed discussion was given by Lapwood [25] for integrals of the t^^pe (2-47). Integrals such as (2-124) wall lead to wave types which can be discussed as follows for the loop L„: 1. it is easy to see that the major contribution to the integrals comes from the neighborhood of the branch points. 55). G{^)e-''' dr - 2ni Z Res (2-124) h = ^.^. 1.\ ^^ J La ^GU)e-"' d^ . a starts as a P wave. . Then 1^ = f 1 ^Jl^. of integrals like (2-121) ISOTKOI'IC JIALK SPACE hi The us(! of . pi/' in seismological nomenclature). since the modulus of exp ( — i^x) decreases rapidly as f moves away from these points. and traverses most of is the path with velocity a.Lff - 2-Ki J2 Res where La and L^ indicate that the path consists of loops lying indefinitely close to the two cuts connecting —ico with k^ and A.JfLa .2W S Re.Lp ^f' m^)e-''' dt - 2-Ki Y. This the reflected wave PS.\J/. factors such as exp [io}(i — x/a)] and exp [zco(^ — x/jS)] will result from the integration along L„ and L^. respectiveh^ Now.(p and p^p and for a shear source. was suggested by Sommerfeld (see Chap. for a compressional source we would have the potentials j. ends as a shear wave. Since the contributions from the infinite arcs are zero. From the type of potential we surmise the velocity of the wave near the detector. j.similar contours for the evaluation and (2-122). t > 0. Ref.<Pa is a wave beginning and ending as a P wave. ^(p and . Res The integrals along the imaginary axis cancel. which also occur in the theory of electromagnetic waves. This fact suggests that.' (2-123) + . provided that the positive of this axis are located and negative parts V on a sheet where no change is required in the definition of some factor in the expression G(^) (such as = ± v/c" — ka in cases considered before) which would affect the integrands. having traveled with velocity a. This would represent a contribution to the reflected P wave (PP 2. respectively. and that wê may associate with these loops waves which have traveled most of the distance from source to detector as compressional and shear waves. For example. to a first approximation valid for large distances. on substituting ^ = k — ir. we have + [ 1 ^ J L^ G{^)e -'' dr .Lb .HOMOGENEOUS AND small circle around each pole.l. 7. j. travels as P along most of the path.(pp /3: is *S. hence they are not represented by the laws of geometric optics. .58 ELASTIC WAVES IN LAYERED MEDIA 3. The first three types of the L^ group do not satisfy a minimum time condition. j. a wave which starts and ends as and P path as 6. finishes as S. This is the surface-generated wave denoted by sPs. Contributions from the loop L^ represent waves traveling over most of the path with velocity 5. This wave apparently be called pS. The contribution from the pole k contains the factor exp [ia)(t — x/cr)] which readily identifies it as the Rayleigh wave discussed earlier. They travel the rules of geometric optics. reflected near the source and . Source Fig. in Figs. This wave appears to be reflected near the detector and will be called sP. 2-12. 2-12 and 2-13.<pff is analogous to 6 in that it starts as S. .\l/p It represents the surface-generated and travels most wave pSp. 4. will 8. is travels primarily as S. These waves are also illustrated in Figs. ends as P. ends as P. Waves from a P source near a free surface. sâ starts and ends as an S wave and travels most of the way with velocity a. minimum time paths and are predictable from Recorder Source Fig. and travels most of the way as S.<pa starts as S. Waves from an SV source near a free surface. is . It occurs for both P and S sources. This the reflected wave SP. of the starts as P. 2-13. These four waves are depicted 2-12 and 2-13.ypp obviously represents the reflected S wave denoted by SS. PS phase from a P source near a free surface. that is. In the range < m < 1. It depends and x and lies in the range < iô < 1. (2-47). X = Vl — + ^4 VaV/^^ — ul we see that a saddle point exists for Wo real. Substituting Uq in (2-128) and we can easily verify that these equations define the 5 is phase PS depicted in Fig. for example. (A-1) we put /(r) = -f-f i^ = P + i<j (2-126) The line of steepest descent is given by cr(A:. Writing f = kû. . is given by (2-128). (s > 0. c > 0). with CO = s — tc. The contribution of the saddle point is obtainable from Eq. we can examine the variation of a for real u. extending the integration from -co to «» and restoring the . 2-14 and that at the free surface. (A-13). 2-14. let us consider the integral }p = —4:1 exp (iocit) I exp i-vh - v'z - i^x) d^ (2-1 25j Referring to Eq. This method (see Appendix A) has been frequently applied to the evaluations of integrals of the type Eqs. z. (2-47).HOMOGENEOUS AND ISOTROPIC HALF SPACE 59 Application of the Method of Steepest Descent. we have u + lVl-u' + l — u (2-127) The function —a or has a maximum — o-q at the saddle point fo given by df{t)/d^ = 0. by exponential functions. on the line between the origin and k^. exp loA t a VI — Uo — = cos 8 (2-129) where Uq (2-129). It contains the factor (2-128) From on h. the angle of incidence of the P wave Free surface Fig. time factor. t) = const. It provides more information about the disturbances at short ranges. Replacing sin kx in Eqs. it must occur when 1 or = xuo + V — /i ul -\- z (2-131) If the intersection occurs to the left of the pole. Eq. (2-130) any value xu of o. The path of steepest descent recuts the line of branch points at or before a point where —a> — o-q. From Eq. This occurs follows that the Rayleigh pulse appears when a/cR > when Uo —> Cr xuo + hVl — — + Z\h2 \^ Uq (2-132) At the free surface 2 = Eqs. 2-7). (2-132) and (2-128) reduce to the condition X a result If > = CrJi Vc (see Ref. of the minimum distance at which the Rayleigh wave . Fig. It the real axis to that of steepest descent. or I ch i[- sw X Vw — 1 X /3^J > If this occurs for Wo + - Vl — Wo + S c = "°] 0. whether one considers the Rayleigh waves to be excited by arrival of first derived by Nakano we introduce the angle 6 sin~^ (cR/a). (2-133) Cr 28 and Sec. 2-15. (2-133) takes the form X > h tan 6. then the contribution of of integration is distorted the pole must be considered when the path from u.60 ELASTIC WAVES IN LAYERED MEDIA To illustrate another application of the method of steepest descent we shall follow Newlands [32] and derive an expression for the minimum distance at which the Rayleigh wave appears. 2-15). (2-133) it may be concluded that the travel times are identical. Interpretation appears. Equation (2-133) has usually been interpreted to mean that the Rayleigh wave does not exist in the interval EP (Fig. ) (2-136) .HOMOGENEOUS AND compressional waves at of the latter point of ISOTltOPlC HALF SPACE 01 P or excited instantaneously at E. the first terms in Eqs. For the discussion of different phenomena these solutions can be generalized for an arbitrary law of time solutions considered in preceding sections are characterized io^t variation. In most problems which we shall consider. Jeffreys [16] suggested the use of a "pulse" represented by a simple Heaviside unit function H{t) changing from zero for for t ^ < to 1 > 0. (pi and xj/i correspond to waves in the first medium resulting from the presence of boundaries. For example. we can put S. The effect of a single impulse of short duration is particularly important. 2-6. (1-40) or (1-41) the angular frequency appears in the exponent only but we can assume that the coefficient A in the former expression is a function of w. u)G((jo)e"'\ In many cases the functions /. z. For such generalizations the Fourier transform has been used in most cases.• are products of the form /(a. The advantage view is apparent if one considers that the disturbance has both shear and compressional components. Now define a function Sd{t) giving the time variation. Generalization for an Arbitrary factor of the form exp Time Variation. or t^ = t — r/a.) = g(c. where (po(. z)(j)^ exp Cr )] since k = u/cr. co) corresponds to the direct compressional wave.c. If we put A = G(a:). (2-119) and (2-120) are of the form F(r. 2)w"~ê"''' occur frequenth'.)F(r. and ^2.it) = -4t= V27r [ -' — di^^y^' d^ (2-134) where the transform g(o:) is given by g(o:) = —^ V27r -'-- f S.h. etc. each of the terms in a solution is found to vary with this parameter and with time as fi(r. If we put t^ =^ t — t/cr. the solutions are constructed from potentials <poe"^\ <Pie'"\ '/'le'"'. or t — r/(3. we shall see that such terms as F(r. z.r.{i)e-"'^ di (2-135) If we put G(cS)<po{0. z). The travel-time curve at of Rayleigh waves is such that extrapolation to zero time gives zero distance from E. In Eq. by the Fourier integral theorem. The steady-state by a time and represent a primary disturbance varying as a simple harmonic function of the time. Then. ^2 represent waves transmitted in the second medium. 26 and 37] we now choose a pulse defined by SM £.. 2-16) Lamb [22. The integral along the infinite half ( vanish because of the factor exp — ioot). * \^ ^ Time Fig. The residues at the poles Tzp yield 9(o}) Le" w ^ <0 w" \ for example. takes the form Then expression (2-137) for /(co) = ^ i co"~' ex-p (=Fco7) + > ii^t^) dco for CO ^ (2-139) which converges.z)e"'''dc (2-137) by superposition. where p and L are arbitrary real and positive constants. 2-16. ^2s-n/2 * COS sm ^^ ^^_ . They represent the waves generated at the bound- Among different functions representing the time variation at the source considered by (Fig. 0) = V \ pL + (2-138) ^ E b . 2-17). This integral can be evaluated r ^n-lg-. G(c.62 terms such as ELASTIC WAVES IN LAYERED MEDIA ^r arise aries.)f(o:)F(r. Assumed initial time variation at the source. provided that n using Euler's formula 0. pp. The Fourier 0) or lower half plane (for transform of (2-138) can readily be obtained from (2-135) by contour integration in the upper (for co < w > 0) of the complex variable circle will t (Fig. Jo COS^ sm ^^ * ^ ^^^^. [ TT doi Jo J-o. We ohiain for (2-l'4'. Ilef. we obtain V . If we therefore make use of the exponential form of (2-140) to generalize the terms in Eqs. p. (2-119) and (2-120) which represent Rayleigh waves. 200). f .)) L^^î^ For n the r(n)ip' + 0""''|exp (inv) - cxp [-in(v + ttj]} f2-l llj < the integral (2-139) diverges but certain information concerning velocities 7 movements may be obtained by working with t and w.scc. Lamb considered a real form of the i.2ii) sin co(i i) dt (2-143) where the summation is for positive co only. Fourier transform and a related function.HI n = l r(|) ( -A K rr" COS u / . 57. The differentiation with respect to can be performed in convergent integrals of the form (2-139). f S.HOMOGENEOUS AND ISOTROPIC HALF SPACE where tan v f33 = ij'p (. Slit) the equations = = TT Jo f dc^ J-00 f Slit) cos coit - i) di (2-142) S2it) . (2-144) .. Fig. 2-17. Integration paths in the complex t plane for oj ^ 0. Instead of (2-134) and (2-135). for example.7r\ = l\/^ = .e. according to Lamb. was also investigated under by Nakano [28. 29]. Sobolev [48]. 2-18. i. Nakano assumed a line source fin the derivation of Eqs. This last problem is considered Hallen [11]. Sin I - 3 2 - u j + (2-145)t Wo = rr-i COS' r-T^ 2n{2rcRyp' KL . 2-19. . In Figs. 2-16 and 2-18 are plotted the pulse S(t) and the hori- FiG. Other Investigations. The surface particle velocities corresponding to passage of the preUminary compressional and of a disturbance in a shear waves are plotted in Fig. Schermann [44]. sometimes called different conditions The propagation haK Lamb's problem. Upper curve zontal and vertical surface displacements go and Wo corresponding to the passage of the Rayleigh waves at a distant point. Naryskina [30. Ground motion. space. and as a limiting case in several papers dealing with propagation in two semi-infinite media. /tt u COS I \4 7 — - . (2-139) and (2-141). / + . 31]. (2-145). 25). is horizontal motion: lower curve is vertical motion.64 ELASTIC WAVES IN LAYERED MEDIA So r-| COS^ 2iJL{2rcnyp i. Following a similar procedure. In the first of the papers quoted above. using the transformation indicated in we can generalize the other terms in (2-119) and initial (2-120) (see Ref. from a distant impulse. 2-7. in the next chapter. note that K is an odd function in k and that K ^ as Re o) ^0. i. Nakano showed that the Raj-leigh waves do not appear at places near the sources. (see Appendix A) was applied in the physical interpretation of the special paths required for this case. 2-19. (6) displacement times 10. A point source of distortional waves was considered by Sakai Following Lamb's method. Receiver Fig.e. (c) velocity. Tatel's model seismograms showing vertical surface motion from a distant surface point impulse.HOMOGENEOUS AND LSOTKOPIC HALF SPACE in the interior of a solid half space Go producing (Ij a longitudinal cylindrical wave and (2) a transverse cylindrical wave. Debye's method of the steepest descent (a) (c) (6) -* —- 10 n sec —> (c) (a) (6) Displacement Displacementx 10 Velocity {d) Velocity X 30 Source iz 4. Nakano [28] used contour integration in a complex plane in order to evaluate the displacements at the free surface. {d) velocity times 30. where the epicentral distance is smaller than (1) . [41]. (a) Displacement..9 cm. This separation is "somewhat arbitrary" (see Ref. however. p. that such separation is possible. transverse. Sobolev [48. However. and Rayleigh waves but these components Nakano forces [29] also considered normal and tangential forces distributed at the surface in an arbitrary way. shear. Sobolev [49] also gave the solution of the two-dimensional problem for arbitrary initial conditions and external forces acting on the boundary of a half space. when there are arbitrary initial conditions given and no external Nonvanishing external forces were by Sobolev as . 28. there is a displacement component of a different nature. 274).66 ELASTIC WAVES IN LAYERED MEDLA. symmetrically about a vertical axis. and Rayleigh waves. In an unlimited homogeneous and isotropic solid there are only compressional and distortional waves but in the case of a half space the Rayleigh wave is produced from body waves of either type when a curved wave front reaches the free surface. and Rayleigh waves are found. Wave-front diagrams are given for these waves. it is assumed. the interpretation given before seems to be a natural one. since on taking different paths when computing the integrals we can obtain different interpretations. besides compressional. This part of Nakano's investigation was developed in order to explain the difficulties in their identification on seismograms at a station whose epicentral distance is not great compared with the depth of the source. appear at about the time required to traverse these distances with velocity Cr. The results obtained for simple harmonic waves were also extended to a more general case where the action at the source may be any function of time. His solution coincides with that of Lamb in that longitudinal. To simplify the problem. 268] that it is possible to resolve a displacement into longitudinal. paying special attention to periodic and displacements repeated in n sectors. where /i Cij = = it depth of source velocity of Rayleigh waves was proved that these waves do not have their full amplitude near these limit distances. 49] applied a new method to the solution of Lamb's problem. under the assumption of a simple harmonic train cannot be separated in the rigorous sense. It was also pointed out by Nakano [28. Following his method. This fact is due to interferences with other kinds of waves. Naryskina [30] found the solution of the three-dimensional problem of propagation of oscillations in a half later considered space forces. At large distances from the epicenter the Rayleigh waves Moreover. Then. p. This wave propagates with the velocity of transverse waves and is the only wave present in case of a transverse force distributed transverse. 375]. In the investigations discussed in the preceding paragraphs. . at the time of arrival of the Rayleigh wave. Each impulse produces a system of waves which may be represented by an equation having a form similar to (2-42). Lamb [23] has also considered the case of an Co impulsive disturbance traveling with a constant velocity direction. say. we can Wo = 1 r" \ -^ A{k) exp [i{oit — kx)] dk + Since these integrals ^ Zir Jo f A{k) exp [f(co^ + kx)i dk (2-146) may become indeterminate. and the expressions for potentials are assumed in the form mentioned at the end of Sec. the direction of x negative. Both components become infinite. while the horizontal component was expressed in terms of elliptic integrals. Pekeris recalculated the solution of Lamb's problem for a surface source and a buried source. Lamb introduces a factor exp (—yt) which represents the effect of a slight dissipative action. Fourier integrals and a special contour in the complex plane are used. and the boundary conditions were usually taken to be independent of time. Hallen problem for arbitrary external forces. Schermann's solution is based on the Cauchy-Fourier method. On of an elastic half space were applying Schermann's method [44] he reduced the problem to a system of integral equations of the Fredholm type. 2-8. 43] for surface normal and shear stresses which depend on the coordinates X and t.J HOMOGENEOUS AND ISOTROPIC HALF SPACE well as 07 [II by Schermann [44] in a three-dimon. In this solution the arrival of the shear wave is marked by a change in slope of the displacements. 1-6. The vertical component could be obtained in a closed form. in a fixed The effect of a traveling disturbance can be obtained by the application of a succession of infinitesimal impulses at equal time intervals. [36] however. The two-dimensional problem was recently discussed by Sauter [42.sional problem. 46]. Assuming in this two-dimensional problem that a concentrated force acts write at the origin. the sources were assumed to have fixed positions with respect to the half space. and the Rayleigh waves in the solution are separated from terms representing longitudinal and transverse waves. Some particular cases of the problem of vibrations due to given displacements at the boundary treated by Shatashvili [45. Assuming a pressure pulse varying hke the Heaviside function. Traveling Disturbance. The method of Smirnov and Sobolev [47] was generalized by Petrashen [37a. Petrashen applied this method also to the problem of wave propagation in a layer overlying a also gave a solution of the two-dimensional semi-infinite solid. f 27r J_a. y being very small. and Eq. co is not assumed to be equal to /cco but.U -j-j exp {ux) exp ( \ Co ) ^-TT U/ for a: > (2-152) Wo for x ^ki < jThe next term dU /dk. . Wo = = is . (2-150).68 ELASTIC WAVES IN LAYERED MEDIA final results are the limits The approached when the coefficient of absorption 7 goes to zero. for those waves whose phase velocity -\- = oi/k equals the velocity Co of the traveling impulse.l_r '^ A{k)e-'"'' dk kco) - tXco - -^^co) 2tJ. Writing k = k k^ and taking the first term (t/ of the expansion. x is replaced ^' an impulse delivered at an earlier time by CqI' — x. < Co Co. We 0. The can be given by (2-146). tor X we have. Since this integral can be evaluated as r" / e'^'' . y - i(cc + ^^ ^^^ In these expressions. take the origin of x at the position of the traveling result of if disturbance at t' i = .^ . form = ^ A{K)e. (2-146) takes the form Wo = TT- A{k) exp \ [icaf — ikicoV — - x) — - yt'\ dk ) I + ^'^ f A{k) exp [icîf + ik(cof x) yt'] dk> dt' (2-147) _ '^'~ J_ 2tJo y r A(k)e"'' dk . A^' — (2-150) difference to the Cojki little The extension of the limits oi k^ to ± oo will make value of Woi in Eq. Eq. 7 — '"'7^^^. where w fcco = [jK- Co j/Ci = - Co)fcit = w{k) U = group velocity of the first integral in (2-148) in the we obtain the most important part Wo._. (2-146) is multiplied by dt' Integrating from = to ^' = oo to obtain the effect of a traveling impulse. dm = if L^ 27re~''"' T" e"" or / dm = r2xe f or x > < (2-151) .. the most important part of the first integral will be due to the root k of the equation CO = kco c (2-149) that is. . The first systematic seismographic recordings from distant earthquakes were made in 1889 by von Rebeur-Paschwitz but it was not until wave 1900 that Oldham [34] recognized the threefold character of the disturbance produced by a distant earthquake and the fact that these three parts correspond to the compressional. Gravity waves in water illustrate the former. when gravity and capillarity both are of the formula (2-154) 5. capillary waves the latter. Equations (2-154) can be successfully applied to the calculations of resistance and to the phenomenon of "dead water. In a second paper on the subject Lamb [24] considered the waves generated by a traveling point source. 2-9. Lamb has also shown IJ how the procedure must be modified in this case if the root of (2-149) also makes = Cq. Experimental Study of Lamb's Problem. 4 is made to the problem of air-coupled The case U = c occurs if c is equal to minimum wave velocity. and or Wo This = I rn Wo = train (2-154) is Lamb's result representing a wave which follows the is less traveling disturbance.. for X > ^ < ^ ." In the latter problem the waves in two superposed liquids are considered (a lajêr of finite depth on a semi-infinite liquid). and Rayleigh waves predicted theoretically. Wo = = A(k) 77 . depending on whether the group velocity or greater than the phase velocity. isotropic solid body and that surface waves may be propagated along a free surface of such a body. taken into account. the second factor unity. shear.— HOMOGENEOUS AND ISOTROPIC HALF SPACE 69 UU > Co. The theoretical studies of Stokes [52] and Rayleigh [39] showed that compressional waves and shear waves may be propagated through a homogeneous.„ _^ Wo U — exp {iKX) exp f yx \ I . 2-18) accounted for man\' of the principal features of seismograms from distant earthquakes for a source not unlike infinite elastic solid resulting . Lamb's region is [22] often neglected solution for the disturbances in a semi- from an impulsive disturbance in a limited one of the most important papers in the literature of wave propagation. Co is 77 — \U for x Col For 7 = 0. Lamb's calculation (Fig. The result is Wo = ± A{k) V2U y = and exp tUx ± - for X ^ (2-155) \dU/dk\ An application Kelvin's surface waves in Chaps. Press. 447-455." Cambridge Uni- versity Press. Geophysics. Kaufman and Roever [19]. vol. vol. F. Trans. Bullen. the presence of large transverse motions in the early part of surface-wave trains and also the long duration and oscillatory character of these trains were so much at variance with the results of the Rayleigh-Lamb theory that there were persistent doubts about the applicability of the calculation. Trace (a) is the vertical displacement.: Pulse Propagation in pp. 1947. and others performed experiments on models. for experimental verification of the Rayleigh-Lamb theory of surface waves in a homogeneous semi-infinite medium. Tatel [53]. Oliver. 3. and trace (d) is this function amplified 30 times. H. M. L. Explosions. 1951. Geophys. In fact. These investigators conclude Lamb adequately explains their experimental results. For example. In Fig. Lawrence: Rayleigh Waves from Small Am. 32. 15. REFERENCES 1. Figure 2-19 with Lamb's theoretical description. 2d ed. limestone. 2-18 is presented Lamb's calculation of the horizontal and vertical ground motion from a distant point impulse applied normal to is Tatel's model seismogram taken with transmitter and vertical-component detector spaced 5 cm apart on the surface of a large block of steel. Northwood and Anderson [33]. 2. : "An Introduction to the Theory of Seismology. Actual seism ograms are considerably more complicated than the idealized one computed by Lamb but it is gradually being shown that most of the complications result from the fact that the earth is a sphere and is layered.70 ELASTIC WAVES IN LAYERED MEDIA that at the focus of an earthquake.. C. Union. Dobrin. B. it is necessary to go to model experiments. amplified and displayed on a cathode-ray oscilloscope whose sweep is triggered by the initial pulse. K. E. It is seen that the essential details of traces (a) and (b) agree the surface. Simon. trace (c) is the vertical velocity of surface particles. In the model study of Lamb's problem a pressure pulse is apphed at a point on the surface or in the interior of an elastic "half space. 1950. . London. Love [26] showed how both of these features resulted from the layering in the earth. the model consists of a block or slab of steel." Actually. The pressure pulse is generated either by a spark or by brief voltage pulses that the theory of applied to a small piezoelectric transducer. 1953. Knopoff [21]. and P. R. Dix. 822-832. or concrete of sufficiently large dimensions to be considered a half space for the distances involved.. using ultrasonic-pulse techniques. Motion generated is by the impulse detected by small piezoelectric transducers. and Ewing [35]. Two Spatial Dimensions. trace (b) is the same with amplification increased by a factor of 10. which affects surface waves far more strongly than it does body waves. pp. 16. 1944. 1-42. SuppL. 307- Kaufman. pp. 1 0. Soc. 2. H. General Theory of Boundary Rayleigh Waves. 10-23. Phil.: The Formation of Love Waves Geophys. vol. 1950. Astron. vol. London. pp. 1951. 18.. Amer. and W. 15. Proc: Srd Congr. vol. Lamb... Seism. A. 321-334.. Fu. Friedlander. 376-3S4. 1949.. vol. W. n. and B. London.. H. vol. 203. Goodier. Y. 34. On the Propagation of Tremors Over the Surface of an Elastic SoHd. Trans. J. Inst. : On 5. pp.. S. Gerlands Beitr. 23. Sec. 14. 1948. A771. N. pp. Phys. in a Two-layer Crust. Soc. 336-350.: Reflection and Refraction of Elastic Waves. J.i72. 1911. Hallen. 1904. E.. 42. Mech.R. Amer. M'.: The and Refraction 1. pp. Mag. Jeffreys: "Methods of Mathematical Physics. Gutenberg. Jeffreys.: "Some Problems of Geodynamics. 13. 6.: Seismic Wave Velocities in Westerly Granite. Japan. Energy Ratio of the Seismic Waves Rcfieeted and Refraoted at a Rockwater Boundary. Knopoff. vol. vol. Nakamura: Notes on the Problems on the Motion of the Surface an Elastic Solid Produced by a Linear Source. Phil. Amer. Sci. G. : [6]. 124-126. Soc. McLachlan. 40-59. Soc. U. 233-326. of Honda.52.. Trans. 1954. 85-102.. G. H. I. 13. Tohoku Univ: Fifth Ser. Chinese : the Propagation of Elastic Waves in a Horizontally Geophys. Science Repts. Energj' Ratio of Reflected and Refracted Seismic Waves. J.. E. Msch. U. On Rayleigh Wave Velocities. 1946. Appl. 5-19. N. Soc: Geophys. Jardetzky. Math. and K. Lamb. vol. H. vol. pp. H. Cambridge University Press. vol.)-'. L. 16. 30. /." 3rd ed.. C. pp. 1952. vol. 125. vol. 12. SHsm. 1916. pp. Quart. Bull." Cambridge University Press. 35. 539-548. London. Jeffreys. at Free Surfaces. 969-973.. 19. Bull. vol. . E. K. 63-100. 20. pp. Y. Seism.. 26. 17. vol. 1946.S. H. 1926. Roever: Laboratory Studies of Transient Elastic Waves. Lamb. 841.. Fu. [6].: Remark on Reflection Critical Reflections of Elastic Waves J. pp. Nakano. Soc (London) A. 28. 21. 329-335. 242. With an Application to : : 24. Publ. pp. Seism. Stratified 7. H.S. 1952. Bishop: On Critical Reflections of Elastic Waves at Free Surfaces. 2. vol. 1947. Appl.: On the Total Reflection of Plane Waves. Mag. II. On Waves Due to a TravelUng Disturbance. pp. pp. Ergin. Studies on Seismic Waves: Geophysics. angew. vol." Cambridge University Press. 22. 1951. vol. Phys. pp. V. 13. Gogoladze. Waves in Superposed Fluids. 386-399. 1931. 23. : 308. Appl. Jeffreys. C. pp. 23. L. 1952. Acad. pp. E. : 1.tsr^ HOMOGENEOUS AND ISOTROPIC HALF SPACE 4. 19. 58-66.. W. vol. Geophys. : 25. vol.: On Rayleigh Waves. die sich von einem zeitlich und ramnhch begrenzten Oberflachenspannungsgebiet in einen isotropen homogenen elastischen Halbraum ausbreiten. vol. : Elastische Schwingungen.. 9. F. Z. 1946. : 1925. The Disturbance Due to a Line Source in a Semi-infinite Elastic Medium. 1-9. Jeffreys. 1916. pp. B. Monthly Notices Roy. Union." Cambridge University Press.. Acoust. 1936. D. 8. the Reflection of a Spherical Sound Wave from an Infinite Plane. Soc (London) A. H. Geophys. Love. pp. Bull. : 11. World Petroleum Congr. 537-545. 27. J. R. : 71 5. Knopoff. Astron. pp. 42. Roy. pp. Some Problems of Medium. Roy. of Elastic Waves. Geophys. Soc. H. 11. 1926.: "The Earth. I. Math. 1946. 1952. Phil.. u. /. pp. "Complex Variable and Operational Calculus with Technical Applications. S. S. Phil. 1279-1280. Ingard. 1954. L. On Wave Patterns Due to a Travelling Disturbance. Amer. and R. H. London. 23. Trans. Lapwood. 1932. angew. 1930. pp.S. T. Sci. H. 41.S. p. Sakai. angew.S. F.S.: On the Propagation of Earthquake Motion to Great Distances. Russian). Bidl. 49. see also C. M. U. Doklady Akad. 39. et Sobolev: Sur une methode nouvelle dans le probleme plan des vibrations elastiques. 1900. 31. Roy. 1934. Geophys.: Der elastische Halbraum bei einer mechanischen Beeinflussung seiner Oberflache (zweidimensionales Problem). 135. Publ.S.: Mathematical Questions in Seismology. 1949.S. Stationary Vibrations Surface of an Elastic Body. Nauk S. 809-811.. Bull. 1955. 1933.R. Inst. : : On 149-153. Press.R.S. U. 239-246. Doklady Akad.S. 202-219.. : Application de la theorie des ondes planes a Inst.R. 38. vol.72 29.S. 1952. Math. Anderson: Model Seismology. 30. Soc. Am. Publ. Petrashen. Nat. Acad. 1954. H. SiHsm. 1932 Russian). pp. The seismic surface pulse. vol. D. 4-11.S. Soc. 2.. vol. London. Nauk S. Sur les vibrations d'un demi-espace aux conditions initiales arbi: 30. pp. Newlands. Two-dimensional Lamb's Problem in Case of an Infinite Elastic Layer Bounded by Parallel Planes. 90.: Some Problems Concerning the Propagations of the Disturbances in and on Semi-infinite Elastic Solid. 1-71. pp. and 29. vol. 1952. Math. Sci.. ELASTIC WAVES IN LAYERED MEDIA. Ewing: Two-dimensional Model Seismology.S. 249-252. R.. Lord: 40. vol. Doklady Akad. E. : On Lamb's Problem of an Elastic Halfspace. Roy. Paris. Publ. 1943. 1900.R. 17.. 45. Z. Nauk S.. 189-348. Shatashvili. Geophysics. Soc. 1-41. S. Trans. vol.: Three Dimensional Problem of Stationary Vibrations Due to Given Displacements at the Boundary of a Medium. Mech. pp. 30. Acad. (London) A. 34. F. 48.A. 1950. p. Math. Northwood. Naryskina. 20. Nakano. J. Publ. 1932. 1949.R. 1953. 1946 (in I. pp. 19. J. Z.S. Soc. Der fiiissige Halbraum bei einer mechanischen Beeinflussung seiner Oberflache (zweidimensionales Problem). 118. 649-652. 8.R. Oldham. Mech. Acad. 43. u. : 37a. Inst. and M. traires. 245.: The Disturbance Due to a Line Source in a Semi-infinite Elastic Medium with a Single Surface Layer.: General Theory of Rayleigh Waves in a Semi-infinite Solid. E. vol. U. pp. F. 1437. 1950.V. Trans. vol. Lamb. Phil. (Tokyo).. 477-493. vol. la solution du probleme de (in H. Pinney. vol. Pekeris. and D. D.S. pp. Acad. On 83. T. Sci. Seism. pp. A. R. Inst. Cambridge University Press. vol. Geophys. Seism. C. Richter. G. 41. Phil. Soc. English summary). pp.. Schermann. Math.S.S.. or Scientific Papers. Sobolev. Rayleigh. 783: 786. 43. • 33.. 42. pp. u. 64. Amer. Proc. 36. U. 1940 (in Russian.. Mag. London Math.: 46. Petrashen. 18. U. S. A. Publ. D. Sci. S&ism. C. 71.S. E. Mag. F. vol.S. pp. On Waves Propagated Along the Plane Surface of an Elastic Solid. .R. Doklady Akad.: A Theorem of Use in Wave Theory. 45. pp. pp. Naryskina. pp.. Seism. 1-10. Sci. L. Sauter. 1885. Oliver. Inst.S. vol. vol. S. Sci. and Phys. 1934. (Tokyo). Acad. Acad. U. Proc. V. 194. 203-215. H. 47. Sauter. 1950. 1951. 64. vol. 44. 30. 35. G. (London) A. 32. Due to Given Displacements at the Nauk S.: Oscillation du demi-espace elastique aux deplacements ou aux forces exterieures donnees a la frontiere.R. pp. 194. S. pp. vol. 469-480. Sci.R. Shatashvili. 376. Smirnov.S. Sîsm. Acad. the Propagation of Tremors over the Plane Surface on an Elastic Solid Produced by an Internal Source. 441-447. 2. 213-308. Calculu." 19] 7-1918. S.stic SoHds. 1951. 56.3. couohf. 8. and G. 53. On the Propagation of Disturbances from Moving Sources. 289-294. Atli acca/L sci. 1952. Research. 373-393. Somigliana.HOMOGENEOUS AND ISOTROPIC HALF SPACE 49. Cambridge Phil.. and of the Equihbrium and Motion of Ela.: 73 f. 47.s. (London) A. "A Course London. vol. 1919. Watson: Cambridge University Press. Sulle onde di Rayleigh. G. Stokes. 40./.. pp... Tatel. k arbitraires. 618-628. Hoc. G. A. pp. 1952.: "Advanced Methuen & Co. vol. London.. pp. Soc. Roy. Stewart. 59. W." 2d ed. : 57. 52. 51.: On the Theories of the Internal Friction of Fluids in Motion. 236-2G6. vol. Hoc. Phil. C. University Press.: Surface Reflection of Earthquake Waves. A. 1849.-onditioris initiales Sur les vibrations d'un dcmiplan ot d'uru. (Moscow). Sobolev. J93. .: "A Treatise on the Theory of Bessel Functions." 4th ed. Camhridge Fhil. E. . Ltd. Whittaker. Torino. pp. pp. N. T. Cambridge 54. pp. vol. H. Proc. London. 55. 1954. G. Sbornik 50. Walters. Geophys. G. Trans. C: Mat. Trans. 53. vol. Walker. N. E. 1951. 109-126. 287-319. Watson. vol. Note on the Nature : of a Seismogram: II. G. of Modern Analysis. 218. we shall discuss several problems concerning the propagation of disturbances in two semi-infinite elastic media in contact at a plane interface. 2-1 that the problem of is essentially simplified by the assumption that all functions involved are independent of one coordinate. and the other only by y. which can be applied only in a few cases. to reflected We shall see in this chapter conditions additional disturbances may arise. Rigid Boundary. We can then discuss two separate groups of displacements. whose axis Ues in the interface. His work was elaborated by other Both earth investigators. between the water and the ocean bottom. Knott [22] seems to have been the first to derive the general equations for reflection and refraction at plane boundaries. In the case of the we are concerned with the interfaces between the atmosphere and the land or water. Reflection and Refraction of Plane section Waves at an Interface. in all details. However. We represent the disturbances produced by a plane P wave incident on the interface from below by the equations obtained from (2-7): (^ = = Ai exp B2 exp [ik{ct — x X -\- az)] hz)] -\- A^ exp [ik{ct — x — az)] . liquids and sohds will be taken into account. this displacement being propagation parallel to the y axis. in general. No disturbance is transmitted to the second medium. We have seen in Sec. Assume common boundary between two media is a plane and that the properties and conditions are such that waves impinging on the boundary from one medium produce no motion of the boundary. 3-1.„_. Under a simple assumption a first approximation that the physical may be made. say y. 20] treatment. and between the different rock layers. one represented by u and w. and in this section we shall make use of Jeffreys' [19. an elementary discussion of reflection and refraction of plane elastic waves will be given.CHAPTER 3 TWO SEMI-INFINITE MEDIA IN CONTACT It is a well-established fact that a disturbance of in one any kind propagating under what In this medium and impinging upon an and refracted waves. before considering this problem. \l/ [ik(ct — — 74 .. interface gives rise. [ik{ct — X — az)] 3-2). 2-1 and Eq. SF-wave Bg-ampl. Fig.1~ B2 2a (3-2) P-wave î-ampl P-wave Aj-ampl. since in the first case. 3-2. SF-wave J5i-ampl. Reflection of P waves at a rigid boundary. Reflection of SV waves at a rigid boundary'. 1 ^ ~ A2 ab ^ . this condition (3 tan' e and tan f given becomes l)(tan' e in Sec. Eqs. SF-wave ''^ Ps-ampl. Then. ab -{- = we obtain. 0. (2-7) take the form (3-4) }J/ = Bi exp [ik(ct — x -\- bz)] -^ Bo exp [ik{ct — x — bz)] P-wave As-ampl. 3-1. using (2-1). a -^ and B2 — > —^ a6 in the second.IN FINITE MEDIA where a b IN CONTACT 75 = = tan tan e f The at 2 directions of interface are depicted in Fig. (2-9) - + 1) = (3-3') If we take an (p incident = A2 exp SV wave (Fig. Fig. w = A. waves corresponding to these terms at any point of the from the condition u = 0.TWO SEMI. . except for grazing and normal incidence. 3-1. .62/^1 ^ 0. \p A reflected distortional wave represented by the function e exists always. There is no reflected P wave if = e tan e tan f = 1 (3-3) If we use the definitions of tan for the case \ = fx. and 2 quantities and by accents quantities referring to transmitted waves. the coefficients c and k must be the same in SViB') SViB^) Fig. Four boundary conditions must be satisfied. 3-3 and 3-4. f are defined in Figs. Reflection of P waves at an interface between two elastic solids.76 ELASTIC WAVES IN LAYERED MEDIA of vanishing displacements at 2 and the condition = (Eqs. solid 1 B2 ah — 1 2& at the interface of (3-5) Any incident wave two elastic- produce compressional and distortional waves in both media. w and the two stresses bodies will. 2-1) leads to Bi a6 + General Equations. requiring continuity of the two components of displacement u. (f we have. Pzz. respectively. tan tan tan e' tan f and e. . f. [ik(ct [ikict = Ai exp 5i exp [ik{ct e f [ik(ct [ik{ct — x X -{- az)] hz)] i^' -\- A2 exp B2 exp B' exp — x x — - az)] hz)] h'z)] (3-6) i// = - + -{- - (3-7) if' = A' 6 a' b' exp - X + a'z)] = [ik(ct - x + (3-8) where a = = = = e'. Indicating by the subscript 1 referring to incident and reflected waves. 3-3. PzT across the interface. in general. in general. Since we assume that the boundary conditions at the interface z = are independent of x and /. + = a' A' - B' . Reflection of <SF waves at an interface between These conditions imply that for real e. p. and we have and refraction immediately follow. we a f. (3-7). indicating phase It will changes..b(B. - {B.TWO SEMI-INFINITE MEDIA the solutions (3-6).) two elastic solids. SViB. IN CONTACT 77 of reflection The elementary laws from the fact that these expressions satisfy appropriate wave equations.. or A. + - A. the four boundary conditions require u at z = B.) u'. A. /3.) - = A' + h'B' a{A. If the displacements and stresses are taken in the form (2-1) and (2-8). as in Sec.-\. p^^ = pL (3-11) (3-12) = 0. a'. than a. Using the methods of Sec. 2-1. e'. 3-4.) B. = p'.. w = w'. 2-1. and /3'. Fig. f the velocity write c must be greater may = n/Iv/iiI c > oi 6 = = ^-1 0/3 c> ^' V a' c> c a' 6' (3-10) = —i\ — <a' h' = -H/1 - ^ c < /3' be recalled that when any of the coefficients defined in (3-10) are imaginary. complex reflection coefficients will occur. a COS e ^' (3-9) cos f cos t cos f' SV{B{) y. and (3-8). c = 00^ A2 Ai For grazing incidence.Â' a A. manner that jS^5^ -^ a sec^ e = c^. For normal incidence. and (3-13) is . p'Ip p Ip e - + ala A' Ai a.)} p'^"{2a'A' + (b" 1)5'} (3-14) By use (1-23). using (3-6). - A. ^% is and -^ 0. is liquid. p'l3"{-{b'' - I) A' + - 2b'B'} (3-13) ju A. let may be expressed in terms of the amplitude of To examine the case where the medium 2 > note that 6 ^ ^ — > oo in such a 0. (3-13).\/Vc~/a 2 1 1 + t a la a' sec p'/p e'. - B. extraneous. For this case Eqs. and Bi = B2 = Slippage occurs at the interface. and the reflection coefficient becomes unity when = a' or cos e = a/a' and = 0.) + {b' - 1)(5: + = B. 2 (3-19) p Ip -\- ala = 0. obtained after a transformation in which made of conditions (3-10) occurs so that either A^ = or By An incident wave of a single type usually = 0. and (1-18). the (3-15) and (3-16) first + = ^ p A' (3-16) directlj^ from the boundary conditions for we equations (3-8). is = piS'. These equations can be derived hquids.)} = P^'{2a(A. .) + 25(5. (3-11) becomes zero. p'/p - a' la ^ p' / p + 2 a'/a p'/p .l/VcVa' + Vc'/a" . and the corresponding tangential stress in (3-14) Liquid-Liquid Interface. (3-12) and (3-13) lead to (3-15) A. Eq. ala' e = 7r/2. From find the reflection and transmission coefficients A.\/cVa" . ^ p'Ip p'/p Ai A' A.78 ELASTIC WAVES IN LAYEEED MEDIA P^'{-(b' - l)Ui + A. and the four amplitude coefficients the incident wave. c = and ^= The e' -1 p ^= tan e (3-20) reflected wave vanishes when = c p\/sec^ t-a la"^ — 1. + VcVa" have by (3-9) cos COS e e' llVc'la - (3-18) 1 where c = a sec = We a a n being the and we have refraction index. A. with a' (hence e') imaginary.e. The refracted wave for this case is given by the first expression in Eqs.2f + âh'] A^ ^ A. therefore. decreasing it. 3-6. The factor representing the phase of the refracted wave can be written in the form regardless of choice of axes and direction of propagation. we may derive for this case the reflection compressional wave in and transmission coefficients for an incident the hquid medium.cya'y Vc'/oc^^^n^ = p tan e ^^^j-. i£<^ M'a[(cV^" .2f + 4. total reflection occurs without change in amphtude and with a phase change of 2e. (. pa'c'/^'' + n'a[{cV^" . pa'cy^" 2pac'(cyi3" . For the case a pa'c'l^" + . A plot of reflection coefficients for given by Eqs. 3-5.a'h'\ A.iVi . wave undergoes a phase change the case a as does the transmitted shear wa^'e.^^ — VC /q: 1 < e < ^ ^ from ^ to / (3-22) The effect (3-9) of the phase shift is to increase the time factor + 2e/co. For < c < ^' < a [7r/2 > e > cos~^ («//3')] total reflection occurs 2e. and ^' decreases reflected compressional exponentially with distance from the interface._. accord- factor in (3-23). from and (3-22) that e is real and. the attendant phase change being given by where (3-27) cote ="'''" pc ^^^^-l -cVc Vi ^-N"-W'-f-V'-^j . since by (3-9) e' is imaginary or zero. in the liquid. Liquid-Solid Interface. (3-19) appears in Fig. from normal incidence to tt as e goes from the critical value. The phase change 2e is plotted as a function of angle of incidence in Fig.2f + The 4a'6'] . (3-8). = cos"^ (oc/a') to 0. One finds A.TWO SEMI-INFINITE MEDIA For the case a' IN CONTACT 79 > c > a. exponentially with the distance from (3-9) that 2e varies from e^r It can be seen from (3-22) and i. for c < a'.. Nevertheless. Following a similar procedure.^ ^^ ^^^ < i^' < c < a a' is negative imaginary. ^"^ ^^^ + M'a[(c7/3" - 2) 2)' + 4a (3-25) b'] K _ A. In addition to the waves discussed in the preceding section. It follows exp [ik(ct - x)] exp (-kVl - câ'^ \z\) (3-23) For c < a' no disturbance is transmitted in the interior of the second medium.. ^ -pa'cV^" + n'a[{cV^" . the formulas represent a disturbance in the second ing to the first medium which.0 0. propagates along the interface.3-17) becomes (3-20 ^1 where p'/p . a transmitted shear wave occurs. / o/ <3| a- oo/ " / ' 00 /lo/ I / / 7 / / / / /o / CNJ / / / 1 j^j / / 1 CO. //V' f/.t <5. J^ / ' / / / ' / 7/? ' ' / // 1 / '''i 1 ^ 1 / / 1 Ad / ^ 1 / 1 1 / y 1 1 > / / / / / 7/ ' / / d /^ o t-l //. y / Jfl^ ^l-^ o lO o CM o o I Cvl vo 1 o o t— 1 80 . 0 3. / a 1 = 2. and <p' and if/' decrease exponen- with distance from the interface. = ^^ + + V' He studied the following three cases: Case 1 «7^' 1.6 1.7 a! la p'Ip 2 3 1.0 3.TWO SEMI.0 3. (3-8). Ergin [10] has computed the square roots of the energy ratios ^2. 3-7 and 3-8. where p' tan ^'~ A. and 77' for reflected P. For a P or *SF wave in the solid incident on the liquid .0 3.0 3.0 His curves for a P wave in the water incident on the solid are shown in Figs. Phase change 2e for various angles of incidence.0. / // . Both tially a' and h' are imaginary in Eqs. transmitted P. respectively.0 60° - V/ / y 40° - 20° - 0° 15° 20° / ^ . I'. .1 1 30° 40° 50° e 60° 70° 80° 90'' Fig.8 3. and f^ ^ ~ V 1 p tan e Ax ^'^ ^ p tan e B' Ai . 3-6. and transmitted *S.IN FINITE MEDIA 180° 1 IN CONTACT 81 160° - / p p - 140° _ 2 n / 7 / 120° - 100° 2€ - 80° - ^ = 3. 5 >^s (6) ^ ^^ 1 20° 40 < 90°-e Fig. 3-8. Square root of the energy ratio for the refracted S.) wave for a P wave incident 82 . 3-7. (a) Square root of the energy ratio for the reflected P wave. .V in water against a solid.1. {After Ergin.5 / / / f ^ i / ^3^ . (After Ergin.) P d^ 1 ^ ^2 3 1 \ 0. - 20° 40° 90°-e Fig.0 J (c) Jl -^ J ^ ^ ^^ / / Cq 0. (6) Square root wave for a P wave incident in water against a of the energy ratio for the refracted solid. Now.75 in Figs.A2 = S.) for a P wave incident Two liquid-solid other sets of reflection and refraction coefficients for the case of a interface can be derived for an incident compressional or distortional wave in the solid.A^ = D.l)W (3-28) o ^ A.0 ^V ^ \ 1 1 \ N^ 1 / ^ 0. 3-9. we obtain A = As An' M&(&' [2m6 = + + n'b{b'' 1) 1) l)]A' + 2hh'{fx - n')B' -I- = Hh' [2m - p. 1.TWO SEMI-INFINITE MEDIA IN CONTACT 83 the corresponding^ (iuantiti(!s for reflected P. 1. Square root of the energy ratio for the reflected in a solid against water. (3-11) and (3-13). 3-12 to 3-14. (3-13). solving Eqs. We note first that Eqs.G.4i -{. _ A + n'jh" 1)]A- + 1) n')B' M(fo' + . {After Ergin. p/p' = {):A. in Figs. and a'/^' = ]. reflected HiV. (3-11). and transmitted P are plotted l.7.5 \ / -AJ / / 1 /i \ \\ / \\ 30° V/ \^y^ 60° \ y ^/ P wave ] 90" 90° -e Fig. with unknowns . (3-12). and (3-14) form two separate groups. B^ + B2 = S' in the second group respectively. B^ — B2 = D' in the first and A^ . Solid-Solid Interface. 3-9 to :^-14 with a/ a' = 0.'{h" - l)]A' 2b-(M + - 6'[2Ai' yi{h' .2. mA' - [m(6^ - 1) - i^'ib" .0 ELASTIC WAVES IN LAYERED MEDIA 3 y =^ # '/ ^% 1 1 ^ ft! 0. Now and refraction coefficients from . A' we have A^ As- + 1) = a'[2M' + ui{b' = Mb' l)]A' - [n(b' - 1) - n'(y' - 1)]5' = 2aa'{tJL' - iJ.) 60° 90= SF wave for a P wave inci- A^ ^ [M(b^ A 1) - n'jb " - l)]A' + b'[2^' + m(6^ - l)]g^ (3-29) fxhih' -f 1) Solving Eqs.)A' + a[2pL + î'{b'^ - 1)]5' and D = S' Ao A' A.. a'W + ^ ~ 2a'{n' m(&' .\)W ixaQ/ + (3-30) 1) = It - ij)A' + [2m + 1) mX^>^' 1)]^' A' n{b' + (3-31) we shall be able to find the reflection was assumed that the incident waves occur in the first medium.5 ll / r 1 j 1 1 30° 90° -e Fig.84 1. (3-12) and (3-14). {After Ergin. 3-10.. Square root of the energy ratio for the reflected dent in a soHd against water. 3-12. Square root of the energy ratio for the refracted in a soUd against water.0 \ > ^ ^^ \ / \ \ bq t 0. {After Ergin.) P wave for a P wave incident 1.5 \ \ \ 1 / \r / f 30° 90° -f Fig.5 ^"^ ^ ^-^ _3__ -^ 2 1 i 90° 30° 60° 90°-e Fig. Square root of the energy ratio for the reflected incident in a soUd against water. 3-11.) 60° 90° SV wave for an ST' wave 85 .0. {After Ergin. 5 f 1 ^^=== ^3 •I >V y 30= ^^ \ / / / / \ \ \ 60= 90° 90°Fig.) P wave for an SV wave -:^^ 0. {After Ergin.) P wave for an SV wave 86 . Square root of the energy ratio for the reflected incident in a solid against water. {After Ergin.5 > I: 20^ 40° 90°Fig. 3-13. 3-14. Square root of the energy ratio for the refracted incident in a solid against water.f f 1.0 3 2 \i mi Ci5 0. h){mi + -\- (3-36) 2(/4W2 {U mjo) {U I U){m2 m^) - h)imi m^) (3-37) where the expressions Eqs.g. . B.. Then these equations take the form A. (3-28) to for the coefficients and m (3-31). (3-32j A2 I 1 A B B' "^ - ~A {U B. Gutenberg [Chap. 2. 2-1) J ^ A| A\ tan tan f e Bl Ai p'i^nt' A'^ p tan e p' tan f^ e B'^ A? p tan Aj ._ W4) + (^2 W3) + â 1 (3-34) -2(^. . . 37]. Muskat and Meres [29] must be taken from developed systematic tables of the reflection and transmission coefficients for the various types of interfaces for application in seismic-reflection surveys. The reflection and refraction of elastic waves at a plane separating two media were also discussed in several other investigations (Schuster [48].. A' J B' ^ B. Ref. .TWO SEMI-INFINITE MEDIA (3-28) to (3-31) if IN CONTACT e. Slichter and Gabriel [50.^ Idimi B' and ^= B^ A. A' B' . . 87 a compressional there i« a single incident wave. The ratios may be obtained from the energy equations (particular cases of which were considered in Sec. 10]. A. Brekhovskikh [4]. Ott [36. (k (k + - l3)(m2 Ai A. Gutenberg gave curves for the square root of the energy ratio of the reflected and transmitted waves for several values of the elastic parameters.. wave {Bi = 0). ih 7^/^ ^3)(W2 ^ 1 ^7 + . Kruger [24]. 51]). h)(m2 ik + -\- h)(m2 + + + + W4) ^4) W4) — -- + ik (I2 (I2 h) + + + + h)(rni + - W3) ms) m^) (3-35) h)(m. A' "^ * A^ ^' 17 " ^ + « + _L + ^4) 4~ ^'' + (^1 + -\- ma) Hence ik = 2 f^ ^' (3-33) ^ + (wo ^ A' m^) ^= 7 ^. CO ^"'g/Â 88 . (After Gutenberg. 0.3 - ' yy ^^ —" v 1 . transmitted P.00 1.1 h 0. subscript 2 to lower layer. reflected 6'F. Fig.^ h ^ ^^^=^/^) 1 .07 1. respectively.10 1. Square roots of ratio of reflected or transmitted to incident energy if incident and reflected or transmitted waves are of different type.3 As 1. t&n t' A''' [ p' tan y f B'"" ]^\ p tan B' "^ p tan an incident *SF wave. . 3-16. 0.036 1.) Subscript 1 refers to upper layer.09 1. The terms on the right-hand side are.TWO for SICMI-INFINITE MJOUIA IN CONTACT 89 an incident P wave tan tan e [ and Bl B.09 1. p' _ for Al B. 3-15 and 3-16 for various cases. // \\ > 0. and transmitted <SF waves. The square roots of the ratios are plotted versus the angle of incidence in Figs.103 0.30 1.14 1.1 w \\ . the energy ratios for the reflected P.4 - (b) '^ SV 1 \ // \\ 0.236 0.07 1.4 (a) P^^ a.286 1.2 -- 1. and (3-8). Reflection of a Pulse Incident beyond the Critical Angle. the time variation in the reflected pulse can be given by a Fourier integral 0^(t) = -~= f G{<^)e"' dw (3-42) According to (3-38) and (3-39). a by /o on (3 9) a = tan e is given by (3-22). we can put G(aj) = CO giu) exp i2e f^ CO (3-43) . changing the sign in the The 2e factor is exponent of (3-21). a a c e . 3-1 we derived expressions for the reflection of simple harmonic plane liquid media. hence the factor 2e/|c«j| is to be added to the time for any value of co. Take the time variation in the incident pulse in the form 0. equivalent to a time increase of v. Had we started with a negative co in Eqs.- = Ai exp ioi{t — I ax -\- az)] (3-38) <Pr = Ax exp iu\ — — ax cos — e az -jr -. — r (3-39) where a.) 90 ELASTIC WAVES IN LAYERED MEDL^ 3-2. giving A 2 = Ai exp ( — ?2e). (3-6). expression (3-6) may be written in the form to a phase change 2e. (3-7). For this case. and by (3-21) the second wave is subjected 2e/|co|. the sign of the imaginary radicals as defined in (3-10) would be reversed. This equation shows that the phase shift determined by the physical constants of the media. In Sec. waves from the interface between two pressional For reflection beyond the critical angle only expressions for the incident and reflected com- waves are involved. — - 1 C a sin e .(t) = -^ g{o)) is f 9(o^)e'"' do: (3-40) where the Fourier transform given by g(co) = :^ f ^ 0^6-'"' dr (3-41) Similarly. = ê-''(? is f^-^dk (3-49) The 1-8. Dover Publications. 91 using the transformation of (3-42) given by Arons and Yennie (3-42) and (3-43) cos 2e From r" . York. "Tables of Functions. Jahnke and F. we introduce -1 (9-.t 1 J -\ mn /« I . = AiG A -^T T > p> yj.. principal part of this integral equal to the negative value of the function Eiiat) (see E. 1943). hence the integral for F{t) < — > — The integrand has a single pole at ^ < —t is zero. Thus (3-48) becomes plane for /. -.TWO SEMI-INFINITIO MEDIA IN CONTAC'T [2]. Thus the expression for the . New Emde." pp.=^ \/2iv L \/27 — / / g(w)e'"' rico + The / / f/fco)*?"" fki (. I CO < = (3-44) ^-~d^= ? 1 for CO in j_c„ CO > where (P denotes the principal value of the integral. The factor of cos 2e is the incident pulse last two integrals may if ?je combined into a the function single one.3-42') J -00 Jo ^. <Pr We obtain (3-45) = 0M) cos 26 + F(0 sin 2e where nO=^/ Take the explosion ^Mdrj e"-''-'' dco(P j '-~d^ ' (3-46) incident pulse to have the form commonly used to represent an T <î(^) < .. despite their difference in sign. io. (J ^ > rt \j (3-47) and integrate with respect to r to obtain To perform and one CO the integration with respect to co we follow in the usual co manner ^ ^ a semicircular contour in the lower half of the complex in the upper half for ^ = ia. <Pr{t) = —0i{t). (After Arons and Yennie. 3-17 and show general agreement with their observations on (/) 2e=150 <7t (g) 2€=180° computed from Fig.(t) cos 2e — K e'^'Eiiat) sin 26 t>0 (3-50) t = -— TV e'""Ei{(jt) sin 2e < This formula yields the obvious results that for 2e = 0. $r{t) = 0i{t). and for 2e = ir. 3-50. Eqs. 3-17.92 ELASTIC WAVES IN LAYERED MEDIA time variation in the reflected pulse becomes M) 0^{t) = <P. Shape of reflected pulse for several values of phase change 2 e. Their results are presented in Fig. Arons and Yennie computed the shape of the reflected pulse for various values of 2e.) . 3-3. The propagation of plane waves in two semi-infinite media separated by a plane Sommerfeld [55 and Chap. 1. and m reflections from a free surface with phase shift tt. must be applied (see Chap. (3-50) for the reflected t pulse does not exclude a disturbance even for < 0.. In this respect our problem is similar to that of the transient response of an ideahzed electrical network. the expression for the reflected pulse becomes are involved.seful beyond the critical angle. one with phase distortion but without amplitude distortion (see Ref.g. e. Fig. for times prior to the application of the incident pulse. 3-1. . Their results are directly related to an important practical problem. 4). that is. 8. The theory given by Arons and Yennie seems in all cases involving pulses reflected It is interesting to note that the second Eq. If a pulse undergoes 0Ût) = (-!)' 0i(t) cos 2n€ — — J -at Ei(<Tt) sin 2n( t > (3-0 1 to be adequate and u. When an impulsive source and a receiver are located in a lower-velocity medium separated by a distance large compared with the distance of either from the plane of contact with the higher-velocity medium. As the distance becomes large compared with the layer thickness. Path of "refraction arrival'' when a2 > ai. In many problems on sound n reflections from the bottom. 4). provided that plane-wave approximations are valid and the reflected waves of various orders are added when they overlap. Jeffreys [20]. that of the "refraction arrival" in seismology of near earthquakes and in seismic-refraction investigations. 55]. 3-18. chap. and other methods. and others have discussed wave propagation for the case where the distance of the point source from the plane interface is finite. 3-18. with phase shift 2e for each reflection.^ J TWO SEMI-INFINITE MEDIA reflection of explosion IN CONTACT 93 sounds in shallow water underlain by unconsolidated transmission in layers multiple reflections sediments. Ref. normal mode calculations. interference between waves of various orders becomes important. interface was discussed in Sec. it is observed that the first disturbance arrives at a time corresponding to propagation along the path shown in Fig. Muskat [28]. Propagation in Two Semi-infinite Media: Point Source. 0. 3-19) may be written in the form (1-72). found terms corresponding to the refraction arrival. 237]. We shall begin the discussion with the most simple case of liquid media. Sommerfeld [Chap. P\ «2. first traversed at the higher velocity ai. h) (see Fig. P2 Fig.94 ELASTIC WAVES IN LAYERED MEDIA observations of travel times it From can be inferred that the part of the 0:2 . It was mentioned in Sec. few of whom took account of the earlier results. This difficulty was first resolved by Jeffreys [20] who. path along the interface is of the path at the lower velocity to the critical angle ^^r arrival. It is interesting to note that this problem was studied by several investigators. 1-6 that an expression for waves emitted by a point source at *S(0. which requires only two potentials. The would be expected for this path from the viewpoint of geometric optics. It the basis of the seismic-refraction method of explora- no energy tion. refraction arrival presented a serious difficulty in that Receiver «i. Ref. It was shown by Joos and Teltow [21] that Sommerf eld's formulas can be transformed to represent the propagation of a disturbance in elastic media. spherical iPo = ^—^. If the time factor exp (icj/) is omitted. 55] developed a method similar to that used by Lamb in the case of a half space. For . the remainder and the angle This is of incidence is equal = sin"^ {aja2). Coordinates for propagation from a point source. the well-known refraction used by Mohorovicic is [27] in 1909 for deducing continental crustal layering. He was concerned with the propagation of electromagnetic waves from a source located at the interface.kj ICa Now in the problem of wave propagation in two semi-infinite media we have to use two different expressions for the displacement potential. using wave theory. 1. Two Liquids. 3-19.= ti {z / J^{kr)k dk V (3-52) Jo where R = Vr" V -\- — hf = Vk' . p. The solution for a dipole located at a certain distance from the interface may be found in his textbook on partial differential equations [55. TWO SEMI-INFINITE the iMEDIA IN CONTACT 95 write medium ^1 {z > / 0) e ^ which contains the source we . the positive direction of z being taken in the medium containing the source.54) With the separation of the factors k/v^ and k/v2.r)/c I'l may / dk Jo + / ^^^^ J'l JJkr)k dk f3-53) Jo assuming that besides the primary disturbance ô represented by the first term there is a second one due to the presence of the boundary. For the second medium {z < 0) we may write an expression similar to the second term in (3-53): <P2 = f ^26" -^^ :(Z-A) J. provided that vi = V/v' V2 - kl^ V2 = Vk. (3-53) and (3-54) can be related to plane-wave reflection and refraction coefficients. = where a.{kr)kdk (3-. Substituting (3-53) e w and the pressure as defined in (1-26) and (1-18) are continuous across the and (3-54) in (3-56) and (3-57) gives (3-58)1 — Qie = ) QiQ — tThe exponential obtain this result.2 propagation of compressional waves in the corresponding medium. The potentials (pi and <p2 must satisfy the boundary conditions at the interface z = 0: d(pi d(p2 (3-56) dz Pi<Pi dz = P2<P2 (3-57) These equations express the fact that the normal displacement interface. the arbitrary functions Qi and Qa in Eqs. {e + Qie = —e ia Q2 (3-59) in the first integral of (3-53) taken as —n{h — z) ior z < h to . These expressions satisf}^ the wave i eciuations V\. It will be seen later how inclusion of these factors in the arbitrary coefficients leads to simple and symmetrical expressions. \'^ at = l.- kl^ (3-55) The is quantities Vi and are taken to have positive real parts. This choice |2| required by the condition of vanishing potential as -^ co . is the velocity of a./o(A. Weyl's formula (1-44) for a spherical wave shows that the primary disturbance represented by the first term in (3-62) can be interpreted as a superposition of plane waves. because the reflection coefficient becomes —1 at this point. It seen that the integrands do not contain poles since. hv^ I (Pi 2 8pi I Jo + V2 sinh v^z e~'"'Jo{kr)k dk + cosh vih + V2 sinh vih e~''"Joikr)k dk V\(8V\ + Vi(8vi V2) - z < h (3-64) z > h (3-65) V2) or ')Jo{kr)k dk (3-66) For z < h the function „. To evaluate <pi.^ . = -^ C /*" | e'^'-'-'' (3-61) The solutions (3-53) and (3-54) now become 5^1 r e-"''-'\Jo{kr)kdk Jn {kr)k dk Jo Vi "^^ ' Jo ' — ^2 ' Jo(fcr)fc d/c .2J where the expressions in brackets have been arranged for interpretation as the reflection and transmission coefficients for plane waves. and writing = 9-2/91. we find Q. according to (3-17) and (3-18).+ (3-63) . . Physically this implies cancellation of the direct wave by the reflected wave at grazing incidence. Jo 5^1 + (3-62) V2_ L^j'i . the remaining integrals in (3-62) and (3-63) may be interpreted as a superposition of reflected and transmitted plane waves. Addition of the two integrals in (3-62) to obtain (3-64) or (3-65) removed the algebraic singularities at A: = ±/Cai or = 0. using in the exponential function in the first term of (3-62) \z — h\ = h — zior z <h and \z — h\ = z — h iov z > h. the limiting case of the Lloyd . A(fc) A(/c) is given by viZ = dvi cosh 7T Pi{dvi —-T —+ V2 -f. Similarly. e (3-67) A is similar expression ior z > P2 h is obtained by interchanging z and h. first add the two integrals.. the sum Pi 8pi + cannot vanish.96 ELASTIC WAVES IN LAYERED MEDIA 5 Solving for Q^ and Q2. i.V2) sinh ViZ -p. from the definitions Pi of and P2.e. Two cosh expressions for ^1 are as follows: v^z _ = . returning to the evaluation of the solutions (3-64) and (3-65). /(co) is analytic. it may also be written in the form S(t) = ^. Then the last relations (for example. = co/ai. If we use notations of operational analysis. the time factor exp (iut) in cp becomes exp [(s — ic)it] — exp {ct + ist). Thus Eq. The branch points at which j'l j/^ ±ka. by the complex variable { — k -\. For c positive the integrand becomes infinite as / -^ <» However. (vSce also Sec. ZTTl [ Jn CO -^^ — la do: = e-''H(t) a> " (3-68) ' where H{t) = for / < 0. Hit) = 1 for ^ > is the Heaviside unit function. cj being real. 3-20. z)e'^^ represents the steady-state solution. where /c„. 2-5) . As in (2-137). In appljdng must be sure that 2. for j ^ 1. pp. TWO SEMI-INFINITE MEDIA mirror effect. For / > equivalent to the infinite semicircle in the upper half plane plus a small circle surrounding the pole w = ia. we may write the solution corresponding to the initial pulse *S(0 defined by (3-68) in the form 27rl Jq CO — to- ' where /(w. ka. When a = 0.. and the contour fi runs from — en — ic to co — ic. 25. 66 and 84)]. the complex f plane (see Sec. (3-47). Q is equivalent to the infinite semicircle in the lower half of the complex the contour f2 is cu plane along which the integrand vanishes. A time variation appropriate for an explosion has been defined in Eq. Now replace of oj. Since w = s — ic is now assumed complex. that the integral converges. (3-69) one r.. f = k^ (s^ — T^ 2ikT — (s^ — c^ — 2isc)/a] be real — c^)/ay and kr = —sc/a'^. Eq. /e Ref. jâ = four-leaved Riemann surface. for the particular time dependence assumed in Eq. the cuts in the must be parts of hyperbolas defined by the last equation and lying as shown in Fig. (3-68) defines the unit function E{t). and that approximations used [see in obtaining /(co) are valid over the contour Lapwood (Chap. IN CONTACT siiall 97 use a method These integrals In evaluating (3~64) and (3~05j we similar to that of Lapwood (1949).ir and consider complex values The signs of vx and v^ have already been specified by the requirements Re > and Re V2 > 0. Re ^/^^ — co^/ay = 0. confining the integrand to a single leaf of the = 0.. 2-5. = co/ois. If c > and / < 0. (3-68). and the cuts forming the boundaries of the chosen leaf are given by Re j'l = and Re V2 = 0.) represent the disturbance corresponding to a source varying with time as exp (ioj/). The definition of the contour fi used in (3-68) involves complex values of the variabl e u = s — ic with c > 0.2) imply that are given by f = ±/c„. Under these -\- and negative or k' — t' < conditions. (3-68) may be readily verified. 98 ELASTIC WA\^S IN LAYERED MEDIA co performed in the complex plane along the °o After these 12 contour results in a solution without a singularity at ^ = substitution preUminary remarks. . Integration paths in the complex f plane for Re w > 0. respectively. in Eq. and for complex co the branch points are not on the real axis. + /^ = f Jo A(k)H'o'\kr)k dk + Jo f A(k)H'o'\kr)k dk (3-70) The function A(A. Fig. There are no other singular points on this axis except those of the Hankel functions at the origin.) has no poles. //o^' and i/o^* vanish along the infinite arcs in the and fourth quadrants. = /. (3-66) and write the sum of two integrals ^. 3-20. Distort the contour of the second integral to the negative imaginary axis together mth the loops £i and £2 lying close to the cuts given by Re I'l =0 and Re P2 = 0. we can distort the path of integration for the two cases Re co > and Re oj < as follows (see Figs. 3-20 and 3-21): Using the fact that first Re OJ > 0. respectively. Distort the contour of the first integral (3-70) to the positive imaginary axis. as mentioned above. we make the integration with respect to . £3. 3-21. Since this contour lies close to for the . For branch hues £' and £" lie in the first quadrant.) . so that axis. Since H^îTr) value of V is = —Ho^\ — iTr) and A(/v) is a function in which the same now used on the positive and negative parts of the imaginary and (3-72) cancel.Jo f K{ir)lli."iiTr)TdT r3-72j I k Fig.= f AU)H^'\tr)^ d^ this case the + f A(f)//i^'(fr)frfr (3-73) Re CO < 0. IN CONTACT '.- f MiT)nn'{iTr)T (It (3-7 Ij h = .TWO SEMI-INFINITK MEDIA Thus /. Integration paths in the complex f plane for Re oj < 0.)'. now be approximated. and by a similar procedure we find <Pr= [ A(f)//<^'(fr)f dr j£' + AU)m'\^r)^dt [ J£" (3-74) CONTRIBUTION FROM Re CO > 0. the first integrals in (3-71) <p. can The Contribution from the loop £2. (l/af V. (6) Loops £' and £" for Re co < f 0. V2 — u^ and f ^f = —u du To determine the sign of Ira on different sides of the branch hne we have the assuraptions discussed in Sec. l/al)~^. 2k. signs are indicated in Fig.. \7r/C„/ L —i[kar — 7) \ / [A(iu) — A{ — iu)] exp ^. (a) Loops £i and £2 for Re co > 0.100 cut ELASTIC WAVES IN LAYERED MEDIA Re V2 V2 = 0. — u du (3-75) 4/J Jo ^l<â. where Im f has its smallest value. With those assumptions.^ A{iu) — Kr — A{ iu) • \ = 2iy\ 2 000 exp 2-^> + '')] Wo^'Hir) exp i[ fr - - . The "" (a) (&) Fig. a positive imaginary component corresponds to the left-hand side of the branch Hne and a negative imaginary component to its right side. We can therefore consider u small and make the following approximations when ^ fr is large: = ka. 3-22. 2-5.. the second integral (3-73) takes the approximate form (/?r' = exp a/—. If = we (r - y -«* = lr + u |7M' ^\h - \T where 7 = — use the first term of the asymptotic expansion of Hq^^ and keep seCond-order approximations in the exponential only. we can put f ^ for points on this line = ±m = kl. 3-22a. where on £2 for both (3-64) and (3-65) A/. From the asymptotic form for Hq^^ we note that the principal contribution of the second integral (3-73) occurs for f close to fc„. (3-76) leads to ^(2) 2iy 8a2r r-2iri 'in /'"'• f Jq Jn co(co — ta) to) N (i^ (3-78) SO that the displacements and wl^^ may be obtained by differentiating r . The path is one of least time. we find *o ^ ^ „ gj^' z. Similarly. we may readily verify that the result is identical to (3-76).Comparing Eqs. kaj is large. r f 2Tn J Jq03 a CO we get _ = 27 ba-iT" -^^^_. [ -^--r '^'^ = 1 «""°^(ô) (3-77) where to = t — R/ai. (3-79) and (3-80) with the expression for the initial disturbance (3-77). For this case the only change is the use of the £" contour and the substitution of Ho^^ for Ho'^\ Following the same procedure. This is precisely the time required for travel from the source to the receiver along the refraction path shown in Fig. the potential <p'/ is given by the second integral of (3-74). 3-18. and we find. We may now form of (3-68). _ = — rfco la -^ 27 bUiT e-"'H{tJ * (3-79) ^1^^^ -^^"'"^(U r 1- (3-80) definite beginning at the time .TWO SEMI-INFINITE MEDIA Using a table of definite integrals. with respect to r (2) and Neglecting the term containing 1 e •-^. again including the factor exp <Pi = J — 0:2 -2 6 (3-76) provided that (z + h)/r is small. . IN CONTACT (3-75j exactly.2 «2 where d^^ = sin~^ (a^/ao). initial displacements is if we The refraction arrival seen to decrease with distance . and the angle of incidence is the critical angle 0c r. the particle velocities and loP' obtained by differentiating (3-79) and (3-80) have the same neglect terms of higher time variation as the order in r and z. These displacements are seen to have a t = z + 7 h = r \- cos "i e„ {z -\r h) f^ (3-81) oix a. 101 we can evaluate {icol). we see that the displacements have the same time variation as the q[^^ initial potential. Similarly. and ^ _/_-!!_ * •^ + 7 For Re CO < 0. generalize these results for an initial impulse having the Applying (3-69) to the source function (3-52). . r (h - zY (3-84) (3-85) . As was discussed in Sec. (I) 1 . This behavior may be thought of as an abrupt jerk followed by an exponential recovery. In the great majority of actual cases the refracting medium is a layer. .iut. . upon integration. 2-5. Oliver. we use the loop shown in CONTRIBUTION FROM £i. 3-22 with v^ = In the vicinity of the branch point we can write Tm /'-. and the effects of its other boundary must be taken into account. we ^"' may write (3-82) as " V^^ ^""p v^ ~ ^"^'' ^ 4/ J (I) |_ ^^^ 2k (II) _| ^^^"^-^ ~ ^^ -iu)]u du (3-83) (III) I (IV) where. Fig. and Ewing [40]. . . 2 2 U^ = 1 fc„.102 as r~^ ELASTIC WAVES IN LAYERED MEDIA The singularity at ^^ = for q[^\ w[^^ and at /o = forgo. An experimental study of this problem was made by Press. - U2 ^rr^ V2 = CO - by the condition Re V2 > 0. — M—IU) = .z) y -" iu[cos u{h + 2) + cos u{h — z)\ CO Using the asymptotic expansion of Ho^\^r). at least over a large part of the frequency spectrum. Wo follows from the assumption of an instantaneous rise time of S{t). first term in (3-73) yields the contribution of the contour V3p^ = f i/<''(fr)[A(zu) - k(-iu)\udu (3-82) where. Then the £1 to (pi. . where 7 = {l/a\ — 1/0:2)"*. Miu) . by (3-67). . the major contribution to the branch line inteas required grals occurs in the vicinity of the branch points.. This theory accounts for the existence of a "refraction arrival" but cannot be used for any quantitative description of field data. For Re co > on the right and left side of the cut. . ? = VkJ. respectively. cos u(h : — z) cos uih : + z) III lU 5 2i St CO -— sin uQi -\. obtain (3-83) except for a change in sign of terms with y. Solutions and (3-87) are unaffected but (3-86) becomes <P\ rb^ r at (/i + " 2)e' • ' for Re CO ^ (3-88) Again generalizing these find. (3-88). results for an initial time variation (3-86) we by applying (3-69) to (3-84)./ TWO SEMI-INFINITE MEDIA (III) IN CONTACT 2biy """ //. In this case we use the asymptotic expan- — ka. = We (3-84). we have 2 r a.. To determine the first term in (3-74) we follow a procedure similar to that used for the contour £i."' and (3-87). (3-92) Jo Oil' tôî^rr) TTfr exp z fr -IT < arg fr < 2ir . 3-2. (3-85). $. + V2 2k. 3-22. 1^ „\^'"'«i (3-86) 5 <P\ S.2 • 2 ly i(h + g)^ ' + ii' 1 i(h - zY (3-87) For Re is CO < we follow the contour as depicted in Fig. Zirl [ \_J â> o: -^—^ — fi do: - icr Jo f -^-^^ o: — da: = J la e-"*'Ei{aL. Im i/. positive to the left the contribution of sion for i/"'(fr)t and negative to the right of the cuts.) * In deriving (3-91) the the integral evaluated by the 5-2 contour has been distorted to the real axis and method used in Sec.) r 'H(U (h (3-90) *{' 28iy r'a. in the first and substitute quadrant r for the vicinity of the branch point = -fca. (3-85). (3-89) =\e-'-H{t. + z)M (3-91) where M ^. Finally. (3-94) for a reflection from the interface. . Ref. Brekhovskikh discussed the problem of reflection of spherical waves. 1. The first and third terms correspond to a pulse decreasing as r'^ and having a shape given by the integral of the initial pulse. The expression for $1^^' contains four terms. the superposition of <p[^^ and <p[^^^ leads to the Lloyd mirror effect at large distances: r 2|^^-. Niessen [33].104 ELASTIC WAVES IN LAYERED MEDIA It is seen that $1^' is a disturbance having the same shape as the initial pulse and beginning abruptly at t = ^+ Ui (î_zA' Zrax (3_93) and depends only on the properties of the first medium. from a plane boundary separating two media. the former term showing the "reflection tail" which is characteristic of three-dimensional wave propagation. He derived expressions for potentials for different values of the refraction index n at various distances from the source. From a plot of e'^^'Êiicrtî) in Fig. Thus $r"' and <l>i^^^ include effects arising from the curvature of the incident wave front.„. The amphtude of this wave depends on the properties of both media and decreases as r"^. the first two beginning at the time for a reflection from the interface. electromagnetic as well as acoustic. the next two beginning at a time corresponding to direct travel from source to receiver. To find given by Sommerfeld Pol [58]. Weyl [Chap. $"' may be interpreted as the direct spherical pulse from source to receiver. van der Norton [34]. 3-17 d it may be seen that $""' does not have a definite beginning but reaches a maximum value at the timxe given in Eq. 64]./„. The second and fourth terms decrease as r~^ and have the same shape as the initial pulse. All terms in $1'^' have amplitudes dependent on the properties of both media. Rudnick [42]. Pekeris [39]. and Brekhovskikh [4]. Since (3-93) represents the first two terms of the expansion of Ula^ for large r. If we omit exp io:t. ^ ^'sin f'^^l-'"^/"^ ^ r XaiT/ ai r (3_g3) Alternative discussions of the reflection of spherical waves have been [55]. The expression 4>i"^ represents a similar It decreases '^\ath distance as r"^ pulse beginning abruptly at i = r + 9l±JL (3_94) equal in magnitude but 180° out of phase with $i^\ This corresponds to the grazing reflection from the interface. at h^ = 1 . there are more special cases graph. and add a solution of the third equation in (3-98) to represent distortional waves. media we assume that the first medium (z > 0) displays the properties of a fluid and the second (z < 0) those of an elastic solid. The function 1^2 represents waves below the interface (z < 0). The formal solutions for . which represent compressional waves propagating in both media.kl ±\F^=±^^ (3-100) where the real part of V2 distortional must again be positive. He gave his results in a form suitable when the two media differ only slightly. We can put yp^= \ Jo S2{k)Jo{kr)e'''''-'''dk (3-99) where Szik) is a function to be determined from the boundary conditions and the coefficient V2 must be chosen as usual to satisfy the wave equation.TWO SEMI-INFINITE MEDIA these potentials. ip2. = = ±je -js= ±Ve . Bnjkhovskikh's formulas should have wider application than the solutions of Ott [36] and Kniger [24] which hold only for angles not too near the angle of total reflection. where a^ = a. In the second pro}>lem concerning two Fluid and Solid Half Spaces. the displacements in the fluid are represented by semi-infinite and in the solid body by Eqs. space we can make use formulas in the preceding problems (3-53) and (3-54). As to the source. Like (3-55). If we assume axial symmetry. we obtain •>'. (1-26) and (1-28). The functions (pi. = y^ P2 (3-98) al and oiz ^2 = = velocities of compressional velocity of distortional waves waves in second medium than in the preceding paraonly compressional weaves. 2 VV. Brekhovskikh series the IN CONTACT 105 made use of Weber's functions and of a terms of which are inversely proportional to the powers of the product of the distance and fre(iuency. and \p2 must be solutions i of the wave equations W. A source in a fluid half space can produce For the case of A source located in the elastic solid medium can emit both compressional of a point source in the fluid half or distortional weaves. (1-28). The second equation of (3-104) leads. is (3-103) and (1-29) when the time factor in (3-101) to taken as exp (zW).{k)Jo{kr)e-'"''-''' dk (3-101)t <p^= I Jo Q.rh at 2 = (3-107) inserting (3-101) to (3-103) in (3-105) to (3-107) (uih).U2/C — 2^2^ 2^2^ V2 2l'2/X2 — P2W = Vi[(2jU2^ — P2W ) — 4:fJL2k\v2] + Plp2^ ^2 (3-111) tSee remark following Eq. S-i will now be determined by three boundary conditions. (1-27). (3-58).J2 = (p.r)2 (3-104) The first equation of (3-104) takes the form dz + dz' + ^^-'^^ ' at 2 dz = (3-105) by (1-26). we obtain S2 exp ( linear equations with respect to Qi exp I'lQie"' Q2 exp ( + v2Q2e~''' + k^Soe""'" — = = I'a/i).106 ELASTIC WAVES IN LAYEKED MEDIA a source of simple harmonic waves are ^j = r ^ j^(^kr)e-"''-'' Jo Vi dk -{- Jo f Q. — vzh): (3-108) ke"'" 2 A/ Pioj Qifi" + (2JU2A: — P2i^')Q2e "'' + 2^2^ ^2*826 " Pico —e (3-109) (3-110) 2^2/12^26 "" + {2jjL2k — p2co)»S2e"' = Thus if we put Vi V2 k P2W A = PiCO 2. the third equation of = m(^ On + ^) = ip. to X2VV2 + 2^2 y^ = Xi VVi at 2 = (3-106) Since no tangential stresses act in a perfect (3-104) gives fluid. by (2-57). These conditions correspond to the continuity of three unknown displacements in the Wy z direction (P„)l and of stresses at the interface {z (Vzr)l = 0): = W2 = (P. . Q-2.{k)Jo{kr)e''''-''' dk (3-102) yP2= I Jo Sîk) J o{kr)e"' ''-''' dk (3-103) The functions Qi. TWO and if SEMI-lNi'JNlTK MKDIA IN CONTACT 107 Ai. the displacements are represented by the equations ^-t + l^l where the potentials ^. from a compressional-wave source located in a is in contact with a second solid half space at 2 = which With the use of definitions similar to those made in the preceding problem. (3-114). We now discuss the problem of propagation of a disturbance solid 0. half space Two Solids. Biot [3] has called attention to the possible importance of Stoneley waves at a liquid-solid interface in connection w4th transmission of elastic waves through oceans.^ (3-116) and are solutions of the wave equations VV.2 (3-117) we put = Vk' . (o-108j to (3-110). = f + ^t^ + «* \pi ^-l.. These integrals by the approximate methods cations. in addition to those discussed in the preceding section. H2 (3-115) By simple transfor lations it may be seen again that (3-113).v'.2k' Aiiik'v. If =\^ vi VV. In addition to the of the preceding sections may be evaluated with several modifiwe must consider A = 0. Ave obtain Qi -= — e Q2 = —e — p^wf — (2fjL2k 'S2 = — — e (3-112) where :^ = - vi[ {2ij. and (3-115) are expressions for reflection and transmission coefficients for plane waves.] pipzcjVz /o_i lov A2 _ A. = 4 ^ 2=1. the branch points /c^2 = ±w//32 and the possible poles at corresponding terms will represent. a transmitted shear wave and waves tied to Later in this section it will be shown that A = corresponds to waves propagating with a velocity less than that of compressional or shear waves in either medium and called Stoneley waves. A2.kl (3-118) . ». The the. — 2t'iPiaj — P2C0 ) A A2 jî A 4:V]V2p}Oo (3-114) /.. branch points kî and ka2. A2 are the other determinants for solving Eqs. Equations (3-101) to (3-103) with (3-112) to (3-115) represent the formal steady-state solutions.interface. Xk)Joikr)e''-'''-''' dk (3-122) Now the four coefficients Qi./. In those problems which we shall discuss. = Q. and.rh (3-124) These equations express the continuity of displacements and stresses at the surface of contact (z = 0) of the two solid media and have the form 2ei dr + |!iL oz dr = ^+ ar \- |jfcL dz or (3_i25) -Tdz + -z-r dz + kff^î = — dz — dz o- + k8.)i = {pj2 (P. - Sê-"''" This system ... therefore. = Jo f Q.)i = (P. = S.e"'' S. a "welded contact" is usually assumed.e""' is O2 = ^26"'=" S.\p2 (3-12bj XiVVi + 2mi ^= oz X2 VV2 + 2m2 ^ dz (3-127) On inserting (3-119) to (3-122) we obtain a system of four linear equations for the functions 0. = r Jo S. we have four conditions which hold at 3 = 0: Qi = q2 Wi = W2 (3-123) (7>..108 ELASTIC WAVES IN LAYERED MEDIA of (3-100) and make use and (3-55). = r^ Jo fl Jo(ilT)e-^'''-'" dk + dk r •'0 Q^{k)Jo{kr)e-''''-'' dk (3-119) (3-120) at2>0 ri^^ = Jo f S^{k)Jo(kr)e-'''''-'' ^. the following expressions may be taken as solutions of (3-117). Si can be chosen to satisfy the boundary conditions.{k)Jo{kr)e''''-'" dk at 2 < (3-121) . the time factor being omitted: ^. Ai. V2 Now the coefficients Q. = = r Jo -Jn(A:r)e-"'^-'" Vi dfc+ ( Jo ^ A Jo{kr)e-'"''^'' dk (3-133) i/'i / — Jo A A ^ Jo{kr)e~'''''~"'' dk (3-134) <P2 = = Jo I -r Joikrje"'''"'' dk (3-135) 4^2 f ^Jo(kr)e''''-'''dk (3-136) .w is = tti 2)U2A. (2)Uifc — P)Co )e " "* 2A. and /-v ^S.ê A' where Ai. the factor exp { — v^h) being separated from them. j'2*S'2 = — A. can be written in the form '^1 -21-1^ (3-132) Q2 = ê-"''-"'' ^2 . By (3-119) to (3-122) and (3-132) the solution of the problem of wave propagation in two solids is represented by the functions • • • . — P2C0 = a2 (3-130) the determinant of (3-129) 1 -1 V2 -v[ -v'. A{k) = vi —k 2/xi/c v[ k (3-131) — tti a2 2/X2A. <^.TWO SEMI-INFINITK MKDIA IN CONTACT 109 (3-129) + 2fXxViQi 2/X2A./iie + 2yu2J'2Q2 — (2Mi/i^"^ — PiwO'S'i + (2/i2/c^ — P2W^)*S'2 = Thus. on putting 2nik — p. The factors Ai/A. A2. and A2 are the determinants formed according to the well-known rule. A2/A have the form of reflection and transmission coefficients for plane simple harmonic waves. (2) along the interface. A. analysis. 3-23 for a compressional interface although the ray cannot be drawn. for and refraction are governed by Snell's law. the propagation paths are minimum-time paths which can be represented by rays. kpi. Different types of waves are determined by a set of branch line integrals corresponding to k = kai. may be considered to travel along a path composed of three parts: (1) source to interface. the coefl&cient of z gives the same information about the third part. of these coefficients Some may assume real values at certain is branch points. and (3) inter- The coefficient of h in the exponential indicates is whether the first part of the path traversed by compressional or shear waves. than that of body waves in either medium.U2 — — Pi) Ml) // 7 2 P2CO .f . These results are summarized in Fig. .„2. In all cases when the exponents are imaginary.110 ELASTIC WA\TS IN LAYERED MEDIA The first term in (3-133) represents the direct compressional wave. The characteristic equation (3-138) in the form given later in this section was investigated for the first time by Stoneley [56].2 Pl<^ I V 2(m2 — 4 Ml)/ \ ^(M2 — Ml) — (^1^2 + I'2'^l) 77 :. reflection. propagated parallel to the apparent exception occurs when two or three consecutive parts of a path are traversed with the same velocity. The wavy lines indicate waves which the paths cannot be drawn. All the v and the resultant wave cannot be represented by rays. in which case the wave is a reflected type confined to one medium. To investigate these waves one has to insert the time factor again and evaluate the integrals in (3-133) to (3-136) by the methods used earher in this section or by some other method. ^^2 and residues corresponding to roots k of the equation ^{k) = (3-137) where A(^-) A(A-) is given by (3-131). or = 4(m2 - Ml)' r\ k C0^(P2 2(. and the velocity along each segment is indicated. Under certain conditions a real root of (3-138) exists corresponding to a velocity of propagation less factors are real. Angles of incidence. while the value of k at the branch point indicates the mode of travel along the interface. corresponding to a wave for which energy An source when a2 > /32 > «i > jSi- The heavy lines represent rays. we may surmise from what has preceded wave types associated with the branch points of Eqs. All other terms in (3-133) to (3-136) represent waves generated in both media by it.: Ti + ViV2ViV2k (3-138) Without carrying out the that each of the (3-133) to (3-136) face to receiver. Source located in lower-velocity medium. amplitude decreases exponentially with distance from the interface fall off as r'^ with distance. The existence of surface waves in an elastic half Stoneley Waves.TWO SEMI-INFINITE MEDIA Its IN CONTACT 111 and can be shown to an interface wave.ft k=u/(3^ \^^ «1 \ «2 k = <a/ai X -ft «2 ^ A= — >- «//3i \ V-^ «2 ^ ft k^H Fig. a2 Waves from > |32 > ai a compressional source near interface between two solid half > |3i. 165-177] investigated the effect of a surface layer on the propagation of Rayleigh waves and discovered another wave of . Love [26. space was shown by Rayleigh in a derivation using plane waves. and Lamb extended these results to cylindrical and spherical waves (see Chap. pp. as would be expected for «^1 ^1 Va •*1 'ffl •'I k = (a/ocz «2 . spaces. 2). 3-23. Other analyses of Eq. that a modified Rayleigh wave with velocity depending on the properties of both media could exist under the stringent condition that shear-wave velocities in the two media were nearly equal. in addition to the ordinary waves with velocity determined only by the properties of the surface layer. On the other hand. W^ exp ik(ct — x). (3-142) 0. Fj..pzAiî 1) Pl + P2} K%i^B. where Ui. - = (3-139) which is equivalent to the characteristic equation (3-138). Stoneley obtained the frequency equation in the form C*{(pi — pz)^ — (P1A2 + P2^*li)(piS2 + P2-B1)} + + 2AV{pA252 . Assuming a solution in the form Ui. /3 medium variables n and = /3i/iS2 . = [l - -. Equation (3-139) was numerically solved by Koppe [23] who concluded that c^/(Si cannot have smaller values than the root of the Rayleigh equation. A2 = 1 - K = with 2(pi/3? - p2^2) = 2(mi - M2) k =c (3-141) and I/. the velocity of surface waves at the interface of two solid media of falls between the velocity the Rayleigh waves and that of transverse waves in There is a region in the plane of where surface waves are impossible. generalized Rayleigh waves are always possible at the interface of a solid and a fluid medium. = If P2 = 0. For ELASTIC WAVES IN LAYERED MEDIA wave lengths short compared with the layer thickness Love found. (3-111). - 1)(A. Stoneley later thoroughly investigated the propagation of this generalized Rayleigh wave (now commonly known as the Stoneley wave). and TFj are functions of z approaching zero at infinite distance from z = 0. Vi. the left-hand side of Eq. (3-139) is the ordinary equation for Rayleigh waves.e. (3-139) were given by Sezawa of greater acoustic density. = Vk^ If Ml kl. Eq.112 the same type. i. (3-139) is replaced by Eq. and the following transformation of variables must be taken into account: A. V2 = kA2 v^ = kB. = fcAi vi = kB. Their velocity is smaller than that of regular Rayleigh waves. This is an equation for the phase velocity c.B. can exist for every value of mi/m2 and pi/p2 which (After Scholte.) 3-4. a spark served as a point source of compressional waves in an acoustic medium consisting of xylol (sound velocity 1. Refraction arrivals observed by Press.) between the curves Stoneley waves A and B. Schlieren photographs revealed all the wave types. 3-24. and Ewing [40] in a . IN CONTACT [47]. and Scholte F'iguros 3-24 and 3-25 represent Scholte's principal results for the conditions under which Stoneley waves may exist. Further Remarks on Waves Generated an Interface. reflected. Asymptote of B P1/P2 Fig. Asymptote of B Fig. 3-25. 113 Cagniard [G].TWO SEMI-INFINITE MEDIA and Kanai [49]. Range of existence of Stoneley waves for Xi/ûi lies = X2/M2 = 1. and refracted waves.600 m/sec). Range of existence of Stoneley waves for Xi = at X2 = °° • (After Scholte. In the preceding paragraphs our study of the propagation of a disturbance in two in semi-infinite media gave terms which represent "refraction arrivals" "Refraction addition to the direct. Oliver.175 m/sec) on NaCl solution (sound velocity 1. in laboratory experiments of arrivals" are readily observed in the field in seismic exploration. They where have also been observed Schmidt [45]. 750 ft/sec) are illustrated in Fig." "waves corresponding to the fourth ray. Direct waves Pi of and refracted waves PiPoPi for a two-dimensional model two layers. according to Thornburgh [57]. 1. 6-1) are used here. to three-dimensional propagation.350 ft/sec) cemented to a thin plate of aluminum (longitudinal velocity 17. may be given by the use of Huygens' which was first applied to these problems by Merten in 1927.500 ft /sec.300 ft. Although longitudinal or plate waves (Sec. A theory including terms representing waves generated at an interface was given by Ott [36] for electromagnetic as well as acoustical waves. the results are completely analogous I I I I I 3." "traveling reflections. (See also Dix [9] and Ansel [1]./sec." etc. simple picture of these waves principle A .) Jardetzky (see Chap. Experimentally and theoretically studied "refraction arrivals" have been variously referred to as "refraction waves. 3-26.4" Plexiglas thickness 1/16" vp= 7. "Flankenwelle." "head waves" (not the Kopfwelle generated by a projectile). Fig. 3-26.114 ELASTIC WAVES IN LAYERED MEDIA laboratory model consisting of a thin plate of Plexiglas (longitudinal velocity 7. Ref. 23) elaborated on this approach to show the physical conditions for the generation of different types of waves in a layered medium. Brass thickness 1/15" i'p= 12. (1-28). p is real and positive.r)^^+ f V\ Jo k dk Q. He used the functions ipi of (3-119) and (3-121) but. Ref. 3-5. 64) could not confirm Sommerfeld's result. Considering axial symmetry. as was asserted by Sommerfeld (see Chap. 3-3. = e"' F. (3-100).- instead of the functions ^j. ioa Since the factor p replaces used in the case of simple harmonic motion. In the case of electromagnetic waves. c = p. other methods of solvwave propagation in elastic media have been applied. and (3-118) take the form vi = = \ik'' + -2 «! V2 = \ k^ \ — +a (3-144) 2- Vl k. he took = —di'i/dr. On making use of a complex refraction index. according to Eq. Ott also found surface waves in the ground when an antenna emits a disturbance in the air. time factor The that the corresponding cuts conditions for cuts in the complex f plane derived in Sec. and exp (io)l) = exp (pt). Cagniard [6] worked with Carson's integral equations [7] and the Laplace transformation. z) U. [ Q2{k)Jo(kr)e''' dk for z (3-147) < (3-148) F.. 55). ing problems of As mentioned above. according to Cagniard. Kriiger [24] could derive the existence of the wave in question by Weyl's method.TWO semi-infinitp: media in contact 115 These terms were obtained by the method of steepest descent appUed to the neighborhood of one of the existing saddle points. Other Investigations. the expressions (3-55).' +1 v'.{k)fÂJo{kr)V'''dk .{kr)e-''' dk ior z (3-145) > (3-146) Fpi f Jo S. put 0. the question of the existence of this wave became the object of numerous papers. After Weyl (Chap.. such a wave is similar to Schmidt's head wave. z) (3-143) where.ik)j^[Jo{kr)y''dk X. 3-3 [Eqs.= = = Jo [ e--'^-'"Jo(A. = Jk' + Again.{k)J. 1.(r. 1. C/.3 = / S. (3-119) to (3-122)] X. 3-3 show now have to be taken on the imaginary axis. Ref. where the real part s = 0. omit the time factor and write the solutions of the second factors in (3-143) in a form similar to that used in Sec. This form of the is a limiting case for the complex exponent w = s — ic used in Sec. According to Ott's conclusion.. = e^'X^r. The variable u having dimensions of reciprocal velocity sions (3-144) as follows: is defined and introduced into the expres- k = pu vi = V\\u -\ — 2 • • • (3-149) We also put «! = = v \\u' + -2 &1 = = yju -\- -^ (3-150) ^2 \\U will +^ Oil ?>2 \/"^ +^ The second variable be determined by the conditions that the expressions (3-145) to (3-148) can be written in the form X. (3-150).r. -d^Pi/dr by (1-28). and (3-143) and the definition Ui = Then (3-154) YAv..r. and ^2 can be determined from the boundary same way as was done previously.- and Fp.(v) dv (3-153) where F{t) represents the action of the source. = \ F'(t Jo - v}A.r.(p. (3-134).r. The fundamental statement in the Cagniard theory is that the functions ^ and f/ giving the displacements in both media can be represented in terms of i^. (3-151) and (3-152) are the Carson integral equations for Ai and S. (3-149). and to introduce the initial time variation by (3-153).z) =p Jo f e-'''B. Si.z)= The -[ f is L|:ô(pur)_ '-''pdu Bessel function Jaipur) the only factor depending on r. 3-3. let us write the making use of Eqs. by methods similar to those used in Sec.r. how the variable v can be introduced. we obtain (3-155) Joikr) =TT Jo f e-"" •=°' " da- .(v.(v) dv U.(v.116 ELASTIC WAVES IN LAYERED MEDIA functions Qi./p and Eqs. Q2. Making use of definition (1-69) of the even function Jo{kr). to obtain Ai and Bi from (3-151) and (3-152). For a given problem.. is the procedure to determine Xp. For further transwill The conditions in the formation we now choose new variables u and v.z) = p f e-'''A.z)dv (3-152) where Ai. Bi are Laplace transforms of X^^lp and Fp. (3-146).z)dv (3-151) Yî{p. In order to see expression for F... = f Jo F'(t - v)B. it is = 0.TWO SEMI-INFINITE MKDIA and. ^^g ^ ^^ (3-156) Jo = V — IT (T Ira / cos 0- rfo- / 'ûe A du (3-100 Jo Jo where. V = iur cos + h\lu^ \ w -\ — ai 2 «i + z\U^ +^ \ Pi [6. J^{kr) ^ dr IN CONTACT 117 = -. Since v is determined by (3-158). and in general the and fourth quadrant also a complex variable.[u(v. It a wave equation. a) |. 0-)]?-"" dv (3-161) . Eq. according to (3-150). a)K{v. a) (3-160) last integral in (3-157) depends also on other variables in (3-158)]. Now. p. 55] To make the radicals in (3-158) uniform functions. u = [u u(v.+ ^ Oil = Vo (3-159) Pi may be proved that. if u is expressed in terms of v and a. in a certain region of the complex variable v. If we compare (3-157) and (1-66). (3-150) to be given positive. real values v>. the can be transformed as follows: -i: f A u(v. Cagniard takes a cut along the imaginary axis in the complex plane of u between the points —i/0i and 61 i/fS^ (when /3i < and /3i < ^2)- We take ai and is real in Eqs. we may easily see that Cagniard's expression (3-157) may be readily derived from that form of solution of play an important part. by their arithmetic values first when u and They are uniform functions in the of the For w u plane. we have Vo = h/ai + z/l3i. therefore. (3-158) The variable a is real but u and are considered as complex variables. (3-158) has a unique root u and that this root is a uniform and holomorphic function of v.r e-'"' Jn IT """ ' cos a da = Thus (3-154) takes the form 2k — Ira ^ TT r/2 r / ^-ikr coa . since for u real the real part of v satisfies the condition (3-159). wave emitted by a source at a distance h from the and distortional refracted waves. (3-165) < < /32 < a2 the wave fronts are represented in computed by Cagniard [6. p. depending on case. starting at a. °o). z) then yielded a set of wave fronts. tortional refracted waves. 3-27. This set was comDifferent distributions of pleted by a similar interpretation of the other three functions B2. Then. we have ^ = £^"£ I dv (3-162) and. the integration path C^ may be replaced by the part of the real axis (vo. therefore. in the form of a reflected distortional wave to reach the height z. very long transformation and evaluation of this so-called transmission coefficient 5i(v. Vo The new of time (for u = 0). substituting in (3-157). the waves generated at an interface. 122] In this figure <S Fig. i.. Since there are no singularities between C^ and the real axis. For the «! example. Wi is the direct compressional interface. v value of cos It may be easily seen that the variable depends on the has dimensions and that the value v = Vo represents the minimum time required by a disturbance in the form of a compressional wave to reach the interface and. using (3-158).118 ELASTIC WAVES IN LAYERED MEDIA integration path C^. after that moment. for the relative values of a^ and /3. In order to see the way in which these wave fronts are determined from W[ and W[ are reflected compressional W^ and W2 are compressional and dis- . wave /3i. Ai. Cagniard proved that the function in (3-161) is an integrable one. we obtain 1 10 —— P = TT / cos a da Ku Ku du e "" dv Jo J dv = -TT r e-^'dv f J to Im du — dv cos a da (3-163) Jo Now from (3-163) and (3-152) the direct conclusion can be drawn that i the solution of Carson's equation for = 1 is given by the expression for V < > Vo BM A = _2 TT f^' Jo Im Ku du Yv (3-164) cos a da for V Vo Cagniard changed the variable of integration from a to u.e. fronts are determined. At u = co v = «> in the first quadrant. The waves denoted by " are the conical waves. A2. r. This and similar figures were for certain particular values of parameters. Eq. A complete set of waves in two solid media produced compressional waves when a2 > 02 > > ft. (After Cagniard. for example. (3-166) can be given the following interpretation: Putting -lo = sm t.TWO SEMI-INFINITE MEDIA the expressions just used. If IN CONTACT 19 wc shall give a short discussion of one example. 3-27. at their Fig. sm z i2 (3-167) and h tan r ii = Vi cos a tan iz = r^ cos a (3-168) in Fig. If we 00 first In (3-164) the root u = Uq of the derivative dv/du is such a pole. where the real parameter I varies from CO J — to the root U satisfies the condition r cos a — hU zlo ll Vl/a. - \/l//3l - = ll (3-166) According to Cagniard. the integrals (8-145) to (8-148) are evaluated by using contour integration in the complex plane. put u = il in (3-158). 3-28 we obtain = r-^ -\- r<i and the relationships represented . the major contribution to those integrals is determined by singularities of the integrands.) by a point source of m poles. was applied to problems discussed in this chapter. which is obviously the time of travel for the wave front just mentioned. The time when this disturbance reaches an observer o- M is found as follows: If = 0. mentioned in Sec. (3-158). Eq. Kupradze and Sobolev [25] had shown earlier that when an elastic-solid medium is . the direction at PM. 2-7. 3-29 (3-170) M{r. we obtain from Fig.120 It is easy to see ELASTIC WA\'ES IN LAYERED MEDIA now that for o- = the condition (3-166) corresponds first to the wave front of a disturbance travehng as a compressional reflected shear from the source S in the direction SP and as a wave wave in Fig. Reduction to the reflected wave PS when o- = 0. 3-29. (3-167) takes the form tic Inn — sm ri sm tap (3-169) 01 Substituting u = iko and r = + r2 in Eq. (3-166). Interpretation of Eq.z) /iao iJ D2/ ho / h / / / / ^1 ^. 3-28. Fig. In a series of investigations the method of Sobolev [54]. 401-448.: Das Impulsfeld der praktischen Seismik 1.: The Interaction of Rayleigh and Stoneley Waves in the Ocean Bottom. Smirnov and Sobolev [52] gave a new method of solution of two. 1950. J. pp. 117-136. Muskat [28]. E. M. pp. 231-237. Paris. Medium of Higher Sound Velocity. pp. {U. no. 6. As mentioned earlier. v. Gogoladze. pp. 1949. 26-32. Gottingen. 3. 13. pp. H. 1948 (in Russian). 1952. Akad. Bromwich. Soc. pp.. A. Cagniard. REFERENCES 1. 15. Soc.. Inst. and D. L. and the displacements were determined to the first approximation. C: "Pulses and New York. C. Ergdnzungshefte angew. Nachr. 10. Gerjuoy. 349-372. Publ. 5... 4. Amer. Bull. 1952. vol. vol.S. R. Seism. II: Discussion of the Physics of Refraction Prospecting. Arons. 1929. A. Seism. 32]. "Reflexion et refraction des ondes seismiques progressives" (These).R. 6.: Total Reflection of Math. Pohlhausen). Reflection of Spherical Waves on the Plane Separation of Two Media. using the methods of Jeffreys and Sommerfeld. 4.: Normal Coordinates in Dynamical Systems.. 4. Amer. Pure Appl. 14. Geophys. 1948 (in Russian).. R. Yennie: Phase Distortion of Acoustic Pulses Obliquely Reflected from a Biot.and three-dimensional problems. I: Fundamentals.: Erdbebenwellen V und VI. Cherry. 18. Geophysics. : pp. 9. B. J. 1953. pp. vol. tions. Proc. larged J.. & Cie. vol. A in . V. vol.similar case of the propagation of a disturbance emitted by a source the solid or liquid was discussed by Naryskina [31. 42. pp. Bull. pp. Springer-Verlag.TWO SEMI-INFINITE MEDIA overlain IN CONTACT 121 Lamb in his by a liquid problem half space the system of oscillations as given is essentially the same but is by supplemented by the existence of new compressionai waves in the liquid medium. vol. G. /. C: Zur Seitenversetzung des totalreflektierten Lichtstrahles. Ergin.S. 238-241. Amer. 1942. 81-101.: Physik. Gerjuoy. Dix.. 1950. vol. . 1442-1449. vol. 12. Ollendorf and K. pp. 6. 22. Sci. Phys. A. 1930. K. pp. Medium of Higher Velocity. 1939. 61] have investigated the problem of a wave generated at the interface of two elastic liquids by Sobolev's method.R.S.S. 42. Phijs. vol. discussed the problem in connection with refraction shooting. 271-278.. Seism. E. 1912. L. : "Elektrische Ausgleichsvorgange edition German und Operatorenrechnung" (enby F. Soc. 4. Wiss. 81-92. Energy Ratio of the Seismic Waves Reflected and Refracted at a Rockwater Boundary. Gutenberg. U. Fragstein. 73. pp. 1939. Acad. vol." Dover Publica- Berlin. 1916. T. Transients in Communication Circuits. B. Rev.: Rayleigh Waves at the Boundary of a Compressible Liquid Medium and a Solid Half space. Carson. Brekhovskikh. 455-482. E. 121206 and 625-675. Ann. 1939. Soc. in graphischer Behandlung.). vol. 8.: Refraction and Reflection of Seismic Waves. Recently Zaicev and Zvolinskii [60. Acoust. : : Gauthier-Villars 7. FA. Tech. They also studied the influence of the ocean on seismic waves. Ansel. 2. 73-91. Waves from a Point Comm. vol. 127. 15. 11. Refraction of Waves from a Point Source into a Source. London Math. and Teltow: Zur Deutung der Knallwellenausbreitung an der Trenn- schicht zweier Medien. 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Reflexion und Brechung von Kugelwellen.of Refraction from a Layer of Finite Thickness. 19. Phil. 31.. Seism. 289-293. 1954. 34. 1953. vol. vol. Publ. Knott. E. Proc.: Theory of Refraction Shooting. Geophysics. Jeffreys. pp. 1930 (in Russian). pp. H. Meres: Reflection and Transmission Coefficients for Plane Waves in Elastic Media. Monthly Notices Roy. 1933. angew. Phys. : . 871-893. Geophysics. 18.R. 321-334. 3. pp. Mech.S. 32. Geophysics. Soc. 443-466. P. Inst. Sci. Z. Ohver. 377-437. A. 28. 40. Jahnke. J. Two Superposed Layers. Physik. pp. Joos. endliche Brechungsindizes. Physik. : 33. Naryskina. Muskat. pp. vol. AI. Ann. 393-404. Z. : 19. Kriiger. Teil IV. pp. Soc. Amer. 1940. C. K. Physik. Ordnung.S. P. 1943.: Das Beben vom 8. G. Geophysics. Physics. On the Theory of Head Waves. Zagreb (Agram). W. H. Kupradze. pp. Die Sattelpunktsmethode : Umgebung eines Pols. 40. 1953. Norton. F. Sci. in 20. pp. H. 149-155. 14-28. Inst. Inst. Koppe. Heelan. Physik. Sobolev: Sur la propagation des ondes elastiques a la surface de separation de deux milieux ayant des proprietes elastiques differentes. Meteorol.S. Press. Muskat. vol. Mem. in der 37. 24. 1933. 1910. 41. J. Geophysics.. angew. Ott. Seism. and other papers. The Propagation of Radio Waves over the Surface of the Earth and in the Upper Atmosphere: I. H. et S. Obs.122 16. M. 1 15-148. 1948. Okt. 123-134. Publ. Sci. u. 43. E. 27. Muskat. 38. IRE. Physik.. A. pp.: Radiation from a Cylindrical Source of Finite Length. 1943. 25. ELASTIC WAVES IN LAYERED MEDIA 17. A. E. 9. M. K. 10. Proc. 1909. 26. V. vol.. A. E. 5. 1367-1387. vol.. 18. A. Love. 1940. 23. Z. Suppl. pp. 22. 1. Naryskina. vol. vol.: "Some Problems of Geodynamics. 1926. 1945. H. 25. C. Pekeris. U. Emng: Seismic Model Stud}. L. Ott.: The Reflection and Refraction Astron. Niessen. vol.: On Compressional Waves 472-481. K. 1899. vol.: Uber die entfernten Raumwellen eines vertikalen Dipolsenders oberhalb einer ebenen Erde von beliebiger Dielektrizitatskonstante und beliebiger Leitfahigkeit. Z.: 39. 30. Publ. 64-97. : Ann. Proc. vol. 22. H. 4. The Propagation of Radio Waves over the Surface of the Earth and in the Upper Atmosphere: II. Norton. : 36. 1909. New York. 1911... Theory of Propagation of Explosive Sound in Shallow Water. 1926. Uber die Schwingungen des festen elastischen Halbraumes der langs der Ebene mit einer elastischen kompressiblen Fliissigkeit grenzt. 48. 28. vol. Acad. vol. Earthquake Research Inst. 1930. 1951 (in Russian). Am. .S. i Geofiz. 1939. Lord: IN CONTACT Dover PublieationB. Bull. Knallwellenausbreitung in Fliissigkeiten und aku. Soc: Geophys. Sci. Phijsik.: Studies in Reflected Seismic vol.5. Z. U. and R. 4.. Phys. vol. 42. C. V. Zvolinskii: Investigations of the the Boundary Between Geograf. 14. Sobolev. v. .. 1936. vol. 52. Soc.: Ser. S4. pp. 1947. K. 61. SUchter. Sommerfeld. 20-39. P. Picht: Ubor die Totalrefie.S.TWO SEMI-INFINITE MEDIA 41. vol. pp. and F. Two Elastic Liquids. 236-266. 53. pp. 554-560. Izvest. vol. pp. 59. 245-266. Physik.: Sur les vibrations d'un demiplan et d'une couche a conditions initiales 55. Assoc. V. Izvest. Geograf. Petroleum Geol. vol. 123 "The Theory of Sound. 38. pp. Z. G. pp." vol. 45.R. 56. 1932. 348-3. pp. 20. 319-326. and K.xion. A. i Geofiz. Rudnick. and V. 47. Recueil math. Van der Pol. S. Smirnov. Stoffe. New York. Amer. Ann. Brechung und Totalrefie. Zaicev. Notices Roy. (London). 41. 44. L. Smirnov. V. of Refraction Waves. R.sten Korpern.: Elastic Waves at the Surface of Separation of Two Solids. Physik..: Theory of the Reflection of the Light from a Point Source by a Finitely Conducting Flat Mirror.S. Aanxsl. U. 1947. Head Waves Generated at Nauk S. pp. vol. 1937. Ann. Z. O. with an Application to Radiotelegraphy. R. vol. 1933.. 29. 40-50. pp. 57. 43.. V. (Moscow). New York. 15.. Rayleigh. P. S6ism.R. Schmidt. O. 2. 49. Monthly 5. 40. 15. K. 891-912. 1. 185-200. pp. Stoneley. 48. Geophys. Zvolinskii: Investigation of the Generated on the Plane Boundary Dividing Two Elastic Axisymmetric Head Wave Media.: "Partial Differential Equations in Physics.S. pp. and Sobolev: Sur une methode nouvelle dans la probleme plan des vibrations elastiques.. 1940. Zaicev.S.: Ser. 1935. vol. B. pp. A.: Schmidt. 19.: The Range of Existence of Rayleigh and Stoneley Waves. (Tokyo). tech. in Reflected Seismic Waves: I. Physics. and A. 46. vol. Thornburgh." Academic Press. Gerlands Beitr. pp. Astron. methode nouvelle a I'etude des vibrations elastiques dans I'espace a symetrie axiale. and N. 843-853. Waves: II. 1933../. Gerlands Beitr. 1-8. Scholte. 194. J. Seism. 1938. Slichter. 19. Proc. O. 416-428. 17. vol. I.R.sti-schen fe.: Wave-front Diagrams in Seismic Interpretation.S. Kling: Zur 407-409. 1934. v. vol. 1939. Sezawa.56. Inst. vol. Sci. vol. 106. Geophysics. Nauk S. 33. 120-126. S. 1933. H.. Kanai: The Range of Possible Existence of Stoneley Waves and Some Related Problems. Akust. Bull. pp. Pich: Ein Beitrag zur Theorie dor Totalrcficxion. Wolf.. Publ.xion in der AkusLik uiid Optik.S. 50. Schuster. A. 30. 1924. 54. 38. Acad.5- Zur Schallausbreitung langs poroser 340.S. vol. Inc. 2. SuppL. Schmidt.. vol.. 19. L. B. Inst. arbitraires.. 1951 (in Russian). le 239-256. Publ. G. Schacfer. pp.: Tho Propagation of an Acoustic Wave along a Boundary. no.. 58. Gabriel: Studies 228-238. B. Akad. Geophys. v.: The Amplitude and Character and N. 60. Roy. L..: pp. Akad. Soc. pp. p. 51.R. L. 1. 1949..: Sur rapplication de 1933. X.- = 2J'=i î^ where of the jth layer. q.CHAPTEE 4 A LAYERED HALF SPACE problem of wave propagation in which a whose thickness is so great. the former is perpendicular to ential equations of motion for the jth layer are we have to The latter is it.- consider only two components of displacement.and three-layered media be discussed in the following pages. Love (Chap. General Equations for an n-layered Elastic Half Space. Stoneley [193] investigated the effect of the ocean on the transmission of Rayleigh waves. The differ- i\ + + 2m. the free surface. d^Wj dz dr dr^ df 124 . Ref. because of the symmetry in the distribution of all quantities involved. obtaining results It is only the exceptional homogeneous layer exists of wider applicability to practical problems. and H„ is = = By is py. Many problems of interest in geophysics and acoustics involve propagation of elastic disturbances in a layered half space.. and My Hj is the thickness we denote the densities (j and elastic constants of the media forming the layers = 1. In the most usual problem the half space is divided into homogeneous and isotropic layers by the 2 co . • • • .) dr^ r dr Qi + dz d Wj dz dr Mi r ->2 a Wj dz dr Pi 12 d 2m.) d^Wj r dWj (4-1) (X. An axial important case again that of a point source. and. planes 2. n). In the present chapter we shall consider cases in which one or more layers are superposed on the half space. 26) gave the first comprehensive treatment of the case of an elastic-solid half space covered by a single solid layer. He calculated the dispersion of Rayleigh waves and showed that a new surface wave having particle motion parallel to the surface and perpendicular to the direction of propagation could exist under suitable conditions. A general discussion for a multilayered half space will also be given. compared with the wavelength. will Problems of this kind dealing with two.- dr dz dz dz dr d2 Wj ~ P.-. that the theories for half space considered in Chaps. 2. 2 and 3 are applicable. and w. 2. treating the bottom as a solid half space. taken parallel to the z axis. 4-1. A LAYERED IIALK SPACE If. = at2 = ^^_^^ If any layer is a perfect fluid. In the preceding chapters (4-1) are satisfied if we have seen that the equations of motion we put ^V. f p.„(^ + fi) set of . ^2. the velocity or displacement must vanish J for all values of time.. the tangential stresses at its boundaries disappear. + 2. atz = 2. + .- (4-2) will hold.n ^ I Wj -^ 0) Wn ^ / as2—>c» (4-0) 0] These conditions are usually replaced by the following: <Pi ^ / I asrôo (Pn ^ I as2—>oo .2. = . we assume that the layers are in "welded contact" at each interface.. ..\ . + i Wi =1^. _. and iPi and i/'. = (7. Thus ' asr—>oo = 1. 1 ^V. In order to adjust the general solution (1-71) of the wave equation to a case invohîng some . The continuity of the normal and tangential stresses (v„l = at the interfaces is X. „.4-3) also assumed. . As to the conditions at infinity... • • • . the conditions g. d(pj .- are solutions of the 2^ wave equations _ 1 ^V. where (4-9) The results for two semi-infinite media indicate that for layered media we can use solutions of the type (3-119) to (3-122). vv. 125 as before. d ( d\p. and we obtain the other boundary conditions P. and the equations concerning the tangential displacements at its boundaries are eliminated.). (4-6) where (p.- and i/'y are displacement potentials defined below. d(p. = rp. It will be seen that these terms correspond to upward and downward traveling waves in the layer. as required by Eqs. Two-layered Liquid Half Space. ^ = 1.kl (4-10) k. The dispersive waves observed by E^ving [35] and Worzel and Ewing [211] in experiments on explosion sounds in shallow water (10 to 20 fathoms) were first interpreted by Pekeris fll6]. we will introduce the spherical wave emitted by the source bj^ adding its potential to the corresponding (p^ or i/'. (4-6). wave propagating from the source with the ai has to For a shear-wave source. For the nth layer of the present problem we > 0. ••• . 01 = (Po + <Pi (4-15) where <Po -w-= K Jo(fcr)e-'^-^'^^ Jo vi (4-16) represents the spherical velocity ai. 4-2.126 ELASTIC WAVES IN LAYERED MEDIA layers of finite thickness. h) can be located in any one of these layers. = V^' . for J ^. for example. 0. we retain both the positive and the negative is values of the coefficient of z in exponent for each layer whose thickness finite. He considered a problem of propagation of a disturbance in . Since this source *S(0. ^ S. rhere ^ ^ — and ve can write. be replaced by jSi and vi by vi in Eq.J^{kr)e-"' '-'' dk r S*Jo{kr)e" " dk (4-12) and ^. retain only factors of the form exp ( — I'z).. a point source of compressional waves is in the first layer. n z-h) 1. -i: î: Q.{kr)e-''' dk + -+ /: QVo{kr)e'' dk (4-11) ^. we put the contribution of a point source located in some layer.J. If. (4-16).2. i/ Omitting the time factor and choosing positive real parts for the i coefficients ^ = Vk' kai 7 kl ' v. = = Jo f QJo{kr)e-""'''''' dk (4-13) yPn Jo f S J o{kr)e- '"'''-''' dk (4-14) Written in the form (4-11) to (4-14) the solutions do not yet include We will assume now that such a source is at a distance z = h from the free surface. Let the first liquid layer be water (ocean) z and the second the "liquid bottom" extending from *S is = H to z = <» . Point source in liquid layer over liquid half apace. (4-22) our results will differ from his in dimensions. and Wj are expressed in terms of the potentials Q. ^. Omittino- Jo f Jo{kr)e-'"''-'' dk + <P2 Jo [ Q^{k)Jo{kr)e""''-'" dk + Jo f Q%k)Joikr)e'' dk (4-21) = Jo f cr)e Q2ik)Jo{k dk <p. and aj. respectively (Fig. 4-1). use of expressions for potentials obtained we write earlier.=^ of the 1 d(p. 4-1. We make = J'l the time factor.-^ dr These potentials are solutions V7 2 ^. J = 1. which would cancel. 1 P2 'z Fig. placed at the depth h in the first liquid layer.2 (4-17)t wave equations J V (Pi = -^ ^To- ^V. ^y". = 1. fSince Pekeris used velocity potentials ./i) «i. A p. In boundary conditions involving pressure Pi{d'<pi/df) by pi<pi. The displacements g. ^1 Pl^l — P2^2 SLl Z = H (4-19) (4-20) = at 2 = we have replaced the pressure In the usual application to simple harmonic waves this involves only the omission of a factor oj". Let the density and the velocity of sound propagation in each r - medium be n S(0.: A LAYERED HALF SPACE 127 a two-layered point source liciuid half space.2 (4-18) The boundary conditions are as in (3-56) and (3-57) ~^ = and "V" (p. Pi "- H <^2. Q\. Q2 may be determined.e + Q\e''' -'''\jo ikr) (4-31) Because of (4-25).e Qê (4-24) + Q. ary conditions (4-19) and (4-20) under the following conditions: ke — ViQxC (4-23) + —e where ^ v. For be written in a single integral with the integrand term — h\ vn z occur for z > h or < k — e . + Q. different expressions for < z < h all three terms h. (fi cosh v^H + 5ii^2 sinh viH) (4-27) = k -. (4-21).e'' + Qê-'' = (4-25) 5. (4-23) to (4-25) 1 1 is A = e — Vie'"' v^H ViB'^" VaC = -2e and we obtain Qi A.h v\ cosh vi{H vi — h) -\- 5)^2 îVo V\ sinh v^{.(H-h) Q*. Then Eqs. this expression takes the form (4-310 — 2Qie''''' sinh vxZ Jo{kr) .H sinh v^H — h) cosh viH + -\- (4-28) Q* = V^ sinh vji -Vi(H-h) e vi 51^2 (4-29) 51^2 cosh viH sinh viH (4-30) Q2 = sinh vxh 2bxke vi cosh vyH \z -|- biV2 sinh v^H As a result of the v'l the function in (4-21) may Eq.128 ELASTIC WAVES IN LAYERED MEDIA The first term in (4-21) represents the direct wave emitted by the source summations of upward and down*S. as in the general solution (4-11). In (4-22) the term with the positive sign in the exponent has been omitted because of the condition (p^-Ôasz^ <» The potentials <pi and <P2 will satisfy the bound.. of three The determinant of Eqs. = ^ P2 (4-26) Thus we have a system Unear equations (4-23) to (4-25) from which the coefficients Qi. The other two terms correspond to ward travehng waves which have been reflected one or more times from the boundaries. (4-21) and (4-22) represent a solution of the problem. cosh ViiH vi — ft) cosh f .. (4-34).^ .2. = Then I the potential cp^ is <p'i = = A{k)Jo{kr) sin v^z dk < 2 < h Jo (p[' \ B{k)Jo{kr) sin v.// + + o. we can consider solutions of the wave equation of the form is if = j F (k) J o{kr) sin vzdk or / Jo F (k) J o{kr) cos vz dk Jo first layer into two parts by the plane represented by two different expressions: We z follow Pekeris in dividing the h. and (4-35) represent the solution of the problem.^ '^"ih fi cosh 1/1(2 - //) - 5.— A LAYERED HALF SPACE 129 Thus. We shall see now that these equations are identical with the solution found by Pekeris [116] using another method. 7 T /I \ sinh vih Jo TTT-i cosh vyH Sii/g + ^T smh i/j/f B^ r. we obtain for I < sinh z < h c f" Jo 1 T n X Hirih tî i/. by (4-21). < 1^1 < H vi can be written as _ o r" Jo i. 25. f^ / kJoikr) V].!/^ 5.1^. _„ '^'~"' (4-35) ' ^ Equations (4-32). To obtain a formal solution it is not necessary to start with the expression (4-21) in which a point source represented by the first term../<.?/^ j/if// — h) ..z dk h <z < H (4-37) = Jo vi f D{k)Jo(kr)e-'''' dk H <z = -iv. r ^/. (4-31').e-'^-"' + QV''"' (4-33) and the potential // <p'/ for h >/. j/. we obtain the potential <P2 iov H < „ dk z < co in the form ^2 = r. v^ = -iv^ (4-38) .<. smh ViH (4-32) For h < z < H the factor depcndhig on 2 in -e-^''^-"^ the iiit(!grand of (4-21) takes the form + z (^.(z - H) „ (4-34) cosh v^H + îjâ sinh vjl Inserting (4-30) in (4-22).z dk Jo (4-36) + and where <P2 Jo \ C{k)Jo{kr) cosv. sirh p. and (4-28). Since the exponential factor depending on the variable z can be expressed in terms of trigonometric or hyperbolic functions. namely.TT — v^ + cosViH •:-sm VxH JT 40^ ^ Thus Eqs.{H — cos v. . B{k). C{k). Then f(r) is given by (4-41). This may such be seen from the Fourier-Bessel integral /(r) = > Jo f Jo{kr)k dk f Jo f{\)Jo{kX)\ d\ (4-42) we choose /(X) to vanish for all but manner that the integral over the plane z = h is unity. . The fact that at this point the liquid above and below moves in opposite directions can be expressed by the equation §^ dz -ML dz = 2 r Jo Jo(. equivalent to (4-32) T (1 X and (4-34). Inserting (4-36) and (4-37) in (4-19). z) . . • dk (4-44) z) j-. _ 2k T'l vi cosPi(iJ' Vi — h) cos Vi ViH + + \ iSiFa sin Fi(i7 i8iV2 sin i8^V2 ldiV2 — h) ViH rA-A'i\ V. There are two new conditions.: • dk (4-45) . where z it becomes infinite in over the plane = his unity. we can solve these equations for the four unknown functions A{k).kr)k dk (4-40) t The function represented by ^ a manner that its integral Ztt Jo f Joikr)k dk (4-41) vanishes everywhere except at r = 0. D(k): Noting that JoikX) I — as X -^ 0.(H : Vi Jo ff ip[' ^ksinvji o /.— 130 ELASTIC WAVES IN LAYERED MEDIA this choice of î' By the boundary condition (4-20) is satisfied.{H + ib^v^ sin _ rj rî — cos _viH „ + —i8iV2 sm viH h) COSPiii _ „ . 0. infinitesimal values of X in such a . and (4-40). h). . or / /(X)27rX dX = 1.4 „ — ^ _ 2A' sin I Vi 1^1 /t sini'iH COS VxH . J'l fThe factor 2 first is used in (4-40) instead of —2 in order to conform with the sign of the integral in (4-21). / 2ksim'ih vi „ _ vi 2biksinvxh COS ViH -\- ifû ib{v2 sin v^H (4-36) take the form. ip[ = <pi' atz = h (4-39) and a condition expressing the discontinuity of the vertical component of velocity at the point source. . : 2 \r" Jo{kr) Jo T>i — Fi — + i8iV2sinv^(H T^: _ „ —iSiPzSmviH + v. This component is everywhere continuous in the plane z = h except at the point *S(0. (4-39). h) ip[ = = n 2 \ r Jo{kr) ^LlHLZi^ — : Ti 7- Fi cosv. •]+. or form representing a spherical wave emitted by the source and another one reflected at the free surface. Pekeris takes the fact that for an impulsive source the integrals representing the rays vanish until the appropriate arrival times.-z-h + 2H) -''^'"'-he-'^^^^'^'"'] + K'[. we can expand the last ratio in Eq.A LAYERED HALF SPACE ]3] Now Eq. (4-45) in a series as follows: Vi cosPi(/f Vi -\-JJjU2sinh(H cos vjl + i5iV2 sin ViH z) — — - z) _i7^. the reflection coefficient (3-17) for plane waves. Using K. representation of One corresponds to a wave propagation by rays and the other by normal modes. The expression (4-45) for ^(' can be written in another form. 4-48) and taking exponential functions instead of the trigonometric functions. K = _ . and those multiplied by K" correspond to n reflections from the bottom. and two of the transformations will be discussed here. This expansion of the integrand into a series of terms which can be interpreted as succes- . Thus the four terms in the coefficient of K represent four rays reflected once.{kr)ê--''''-'' dk l-i'i - Jo f J. expressions (4-45) and (4-46) become All those intof^rals redu(*e to a ^[' = [ Jo J. In support of this identification. 1 + i Ke -i2VAH-z) ' _|_ Ke~''^'"" = e-"-[l + Ke-'''''"-''][l Ke-'^'" + K\-'--" + • • •] (4-49) Therefore k dk ^1 / Jo{kr) -rr^^l {[g - e '' '] Jo + K[e -iVi(..{kr)—e-'''''''' dk iJ'i = <p.} (4-50) As we have seen surface z earlier. (4-37) becomes equivalent to (4-35). Now. This interpretation had been used by Sezawa [160] for the case of a liquid layer over a rigid bottom. For these conditions.. the successive terms in (4-50) can be identified with rays reflected a certain number of times from the bottom. according to Pekeris. if we assume that 5i —> 1 and F^ -^ Vi. (4-47j Equations (4-44) to (4-46) can be transformed in different ways. the first represent the direct wave (or ray) two terms of the right-side member and the ray reflected from the free = 0._' . (4-52) = Jo hdt -\- Je hd^ + Joae hdt - 2Tri Y. Res I^ (4-53) provided that the complex poles are not located on the permissible sheet of the Riemann surface. V2) the multiplier of 2Jo{kr)k dk in any of the integrals and (4-35). as cuts. F{vi. by integrals along the imaginary axis. To evaluate tV plane is these integrals. as will be shown later. The contours and the 4-2 and 4-3. and by the <p residues. shown in Figs. so that they overlap. or (4-44) to (4-46). (4-35).)kdk+ f Hl. Integrals of the form (4-51) and (4-52) are improper. contour integration in the complex f = fc used. The integrands determinant (4-27) or at the roots of become infinite at the zeros of the the denominator in (4-32). 2-5. and travel times for successive pulses become nearly equal. (4-34). 4-2 and 4-3. Thus. In such cases pulses become prolonged. In the present work. Denoting by (4-32). our principal interest is in cases where the horizontal distance greatly exceeds the thickness of the layer. a more lies in suitable and direct method for these cases evaluation of the original integrals by methods of contour integration used in the preceding chapters. As would be expected. More precision is required in the definition of the integral (4-51) since . from Eq. are chosen with particular attention to functions involved. The integrals along the real axis may be replaced. The meaning chosen for improper integrals such as (4-51) is + the limit of the integral along a path like that indicated by the continuous lines OM in Figs. Newlands [105] used this method and gave detailed descriptions of the pulses.'\kr)F{v„v.132 ELASTIC WAVES IN LAYERED MEDIA is sively reflected pulses due to Bromwich [12].{kr)F(p„v2)kdk /: or <p= f H'o'\kr)Fiv„v. we can write (4-51) J. using Cauchy's theorem. by branch line integrals. was done in Sec. as before. are Hankel functions and second kind.)hdk Jo Jo = [ Jo hdk i/o^' -\- f Jo hdk of the first (4-52) where Hq^^ and respectively. This definition has the advantage over the if definition of (4-51) as identical with its principal part in that the value of the integral is unchanged the poles are displaced infinitesimal distances into the fourth quadrant. (4-34). A LAYKRED HALF SPACE the part of the real axis from the origin to the branch points the cut corresponding to the condition 133 is a part of chosen in Sec. 2-5. Re j/, = 0, Moreover, substituting IIank(!l finiciions instead of the Jiessel function, is we note that the origin a logarithmic singularity for both of those functions. It is a regular point, however, for Jq. m' (1) O \M Fig. 4-2. Integration path for integral containing Ho^^\ M ; k m^' Jl.^Fig. 4-3. Integration path for integral containing /fo"'. 134 ELASTIC WAVES IN LAYERED MEDIA As to the type of the Riemann surface which can be used to transform the integrals by Cauchy's theorem, it seems at first that there are two radicals introduced and, therefore, the surface should be four-leaved. If the integrals for the first layer are written in the form (4-21), where as a separate term, both integrands and V2- The solution seems to contain terms with contributions from both branch line integrals corresponding to branch points k^^ and fc„,. However, if the direct wave is combined with the reflected wave, as in (4-32) (4-34) and (4-35) then all integrands are even-valued functions of vi and the corresponding branch line integrals vanish. It appears that the disturbances corresponding to the branch point ka, vanish through destructive interference between the direct wave and waves arising from reflections at the surface and at the interface. It will be proved in Sec. 4-8 that in the general problem of this kind where there are parallel layers overlying an elastic half space only the branch points corresponding to the latter medium determine the Riemann surface. Thus, in the problem of a liquid layer overlying a liquid half the direct wave from the source appears vi are mixed- valued functions of , , , space now considered, the . Riemann surface is two-leaved, as represented in Figs. 4-2 and 4-3 The bra nch point A of these figures corresponds to the factor V2 = ± Vt^ — ^l,, which makes F{v^, va) a two-valued function of V2. If we proceed as in Sec. 2-5, the cut given by the condition Re vz = begins at k„2, runs to the origin along the real axis, and then goes to — too along the negative imaginary axis. Im V2 is positive in the first quadrant and negative in the fourth quadrant. Re V2 is positive in both quadrants by the choice of the Riemann surface. We shall denote by the and Im ^2 > or Im vo < 0, respectively. symbol vX or vi that Re I'a = By use of the relation Ho^\^r) = — ii/'o^'( — fr), the first integral in (4-53) can now be written as f Jo Hl,'\irr)F{v,,v\)iTi dr = - Jo f Hl,'\iTr)F{v,, v\)iTi dr which combines readily with the second term of (4-53). Thus ^= f Hi'\irr)[F{v„ vi) - F{v„ v;)]fn dr + = J £. Jka f H'o'\kr)[F{v,, vi) - F{v„ v'-2)]k dk - 2Tri 2 Res (h) (4-54) f - 27rz X) Res (h) is The solution (4-54) composed of a sum of residues corresponding to A LAYERED HALF SPACE real roots ] 3o and a branch line integral along Jb. 'I'he integrand of the second integral in (4-52) has the form /, = k„ H^"{kr) ^ this f4-55; The residue at a simple pole for an integrand of form is given by Res For solutions of the (7.) = H'^\k,x)P^^ /c„ (4-56) form (4-44) to (4-46) is the root of the equation N{k) = vi cosViH + idiV^sinViH = (4-57) with this condition ,,, . sin Pi^ sin Fi/i ^("") = sin VM '' (*-58) N'(Kn) = —^—^ _ f j£ [ViH - sin ViH cos v,H - 5? sin^ v^H tan ViH] (4-59) By , (4-44), (4-45), and (4-54) to ^li Y^ (4-59), the potential for the upper layer is _ ff _ Hl^~\K„r)ViH sin Vih sin v^z H n viH — sin viH cos ViH — 5; sin" v^H tan ViH (4-60) In treating an improper integral of the type (4-51) chose to define the integral as identical with its Lamb (Chap. 2, Ref. 22) principal value. His final result for the residues represented standing waves. improper integrals (4-52) differs is, and —iri lines in Figs. 4-2 and 4-3 from the small indentations above the poles. These terms correspond to the free waves added by Lamb to his result to produce outward propagation of the waves [Jardetzky [72]; see also ^Miittaker and Watson (Chap. 1, Ref. 66, p. 117)]. Returning to a discussion of the roots of the period equation (4-57), it will be recalled that we assumed the absence of complex roots on the permissible sheet of the Riemann surface in writing Eq. (4-53). The OM ^ Res His definition of the from ours by the amounts — xz" Res (/O ^ (1 2), that by the contribution to the integrals along the equation Ai is = ka, jî cosh viH + 81V2 sinh v^H = V,- (4-61) identical with (4-57) because of the definitions of = —iv,V2 given in (4-38). For a^ < 0:2, < ka„ and for k > k^, both Pi and are real and 136 positive. ELASTIC WAVES IN LAYERED MEDIA The left-hand for member k of (4-61) cannot vanish for fc„, < A; < oo and similarly in the interval kî < < k„ < < = k^^. Thus the real roots k„ of (4-61) must he k^,. To locate complex roots, we put Hv, HVr - fca. = = Pi Hv2 = i^VT' - kâ._ Pa + + ki iq2 /4_g2) and pa > 0. The coefficients q^ and ga are positive in the first with pi > quadrant (Fig. 2-7) and negative in the fourth. For w real, /c„, and fc«, are also real, and therefore the preceding equations yield the condition Pi9i = P2g2 real (4-63) Substituting parts, (4-62) in (4-61) and separating the and imaginary we obtain two equations Pi + 5iP2 tanh pi = tan îf^^^tanhpi \ + Sig^ Pi (4-64) 2^ + Pi ^lOa tanh pi obtain + tan qîyi tanh pi + b{p^ = EHminating tan qi, we Pi(pi q2ip2 + îP2 p, tanh pi) tanh + ^ _ q2(p2 + pi(pi S^pi tanh pQ 5,pi) tanh p^ + (4-65) 5,^2) Because of the convention of signs mentioned above, this condition cannot be satisfied in the first or fourth quadrant. Therefore we must draw the conclusion that the complex roots of the period equation (4-61) cannot be located on the permissible sheet of the Riemann surface as defined in the preceding sections. The same conclusion can be reached if we write Eq. (4-61) in the form 5i tanh (p, + qî) = -^L^-iil P2 + (4-66) q2^ and separate the real and imaginary 7)1(1 parts. Two PiPa ps equations result: rA-r7\ yt ^l) 5i tanh + tan' gQ 1i,l2i2 tanh tan + 1 _ — pi q^ + 9ig2 2.2 + ^2 gitan 1 gi(l - tanh' pQ ^ qi Pi^z p' - p2gi ql + tanh' pi tan' satisfied + (4-68) and Eq. (4-67) cannot be with the convention of signs used earlier. We can easily see that there are no roots of the period equation (4-61) A LAYEKKD HALF SPACE 137 on the imaginary axis. If wc put f = — im, this equation takes the form Vm'T^!, which has no COS UVm' + kl. + i8i Vm' + m. kl, sin H Vm' + fc^, = ('4-f/^) real roots Schermann [151] investigated roots of the period equation of a lirjuid In principle, his results for this more complicated case should be the same as those for the present problem. layer over a semi-infinite solid. Schermann found a of finite number fall of real roots complex roots. From the discussion just given, and an infinite number we must conclude that involving contour these com.plex roots do not on the permissible Riemann sheet. Discussion of Solutions. integration, In many investigations branch line integrals are interpreted as terms which are The asymptotic behavior of such integrals by different methods (see, for example. Sees. 2-5 and 3-3). The branch line integral in Eq. (4-60) corresponds to a wave traversing the refraction path shown in Fig. 3-18. Schermann has shown that in the two-dimensional case a branch line integral is of the order r~'. The am.plitudes of waves corresponding nonessential at large distances. and their physical interpretation can be discussed to these integrals in three dimensions (see Sec. 3-3) diminish as r~^. shall, therefore, discuss in greater detail We the more important part of the solution (4-60) given is less by the residues, for which the decrease wdth distance rapid. (pi', having the same form (4-60). Thus these terms represent waves in the whole layer (0, H). Similar considerations prove that the part of (P2 in Eq. (4-46) due to residues is given by It may be seen that the expression (4-45) for in Eq. (4-44), also takes the poles as (p[ <P2R — 27rt5i TJ H 2^ viH ^ Hr(Knr)viHs'mVihsmv,H e''''^'""^ — sin viH cos viH — 5i sin' ViH tan viH forz > H (4-70) An of the for discussion can approximation of the residues in Eqs. (4-60) and (4-70) convenient be obtained if we make use of the asymptotic expansion Hankel function TT m'\Knr) (7r/c„r (4-71) Let viH = X vl^'H = -HVkl, F(a:„) kI = x^ vt' = - Vkl^ - ^ sm" .r„ (4-72) and = x„ — sm -. ^ x„ cos x„ — ^^^; 8'i tan (4-73) .t„ 138 ELASTIC WAVES IN LAYERED MEDIA if Then, we use —i = exp ( — t7r/2) and replace the time factor exp {iwt), the residue terms in Eqs. (4-60) and (4-70) yield the terms îR = \— 2^ ~7= exp -nA H\r 77 i[o:t — Kj- — - V(Xr) sin ^ " sin '-^ f or < 2 < i7 (4-74) 26i <P2R /27r H . 7^ ^ ^ ^ '"P L^"("' '"' ~ - i)J H)] F(x„) sin sin x„ exp [-ivl''\z foYz>H (4-75) Each term in Eqs. (4-74) and (4-75) will be called a "normal mode," the mode being characterized by the subscript n. This subscript indicates that the corresponding quantity is to be evaluated at fc = c„ /c„, where co. k„ are the roots of the period equation (4-57) or (4-61) for a given exponential factor it From the can be seen that the phase velocity for the nth root can be expressed by c„ = c, (4-76) The period equation (4-57) defines an implicit relationship between k, w, any two of the three variables the form tan H{kl^ and as may be seen by writing it in 5, - k^ = -f|Jt=/yi real, (4-77) For ka, < k < kc^, all terms are and real roots k„ exist. In Fig. 4-4 are plotted the curves representing the right and left sides of Eq. (4-77) as functions of the parameter H{k\^ circles define the values of k„ — k^)\ The intersections shown by which are real roots of the period equation. finite value of co the number of real roots is the number H{k\^ — /cIjV't + 1, the trivial root at the origin being included. Thus there is a finite number of poles of the integrands of (4-44), (4-45), and (4-46) on the real axis. It is seen that for any nearest integer to the It is particularly instructive to write the period equation in c terms of and k as follows: 1 ^ i~2 / 2 r~2 / \ / tan kHy ik~ - 1 = P2 Pi \Q!i -V V'- c 12 a (4-78) A LAYERED HALF SPACE 139 «2 When Eq. (4-77) is written in > c > «!. This last ccjuation this form, real roots correspond to the case is an implicit relation between the frequency (4-79; and the phase velocity that the frequency is c. For a given value of phase velocity it is seen a multiple-valued function of phase velocity, each 3i tan Hyfkl~-k Fig. 4-4. Method of determining real roots of Eq. (4-77). value belonging to a different best treated numerically mode of propagation. of c This equation is by choosing a value is and calculating the corresponding value kH. It evident that as c/ai —» 0:2/0:1 kH -> (2n - 1) - 1 (4-80) This corresponds to the cutoff frequency of each mode, limit of possible frequency for this mode. As c/a^ —> 1, i.e., the lower kH (4-81) 140 ELASTIC WAVES IN LAYERED MEDIA it is and the seen that propagation at the phase velocity c = a^ occurs in each of mode at the highest frequencies. = 2Tr/l, large wave number A; More precisely, from the definition c values of kH I correspond to large values <<^ of the dimensionless For a2/«i > c/tti > parameter H/l. Thus 1 the range of k„H for is H, ^ ai for all modes. such that n — -JTT < KnH- I < n-K (4-82) As n 00 , that is, for the higher modes. - 1, and the modes form a harmonic series. It can be shown that the period equation expresses the condition of constructive interference between plane waves undergoing multiple reflection in the liquid layer at angles of incidence beyond the critical angle d^r = sin"^ (cui/aa). In Fig. 4-5 ADEF represents a portion of the G N D BC/GN = Iq/1. Fig. 4-5. Interpretation of frequencj^ equation path of a plane wave which has been totally reflected at the bottom with a phase change 2€ and reflected at the free surface with a phase change of — TT. As the wave front, shown by the dashed line, moves a distance BC = in unit time, it traces the distance is GN in the horizontal direction, where k GN = BC c = ai/sin 6 the phase velocity defined earlier. For this representation the 2Tr sin d/lo, wave number k may be defined by the equation Iq is the path where and I = the wavelength measured in the layer along /o/sin 6 is the wavelength measured in the hori- wave front GBE is to interfere constructively with the coincident wave front which has traversed the additional path zontal direction. If the BDE, it is required that 27r BDE - 2eie) -\- X = 2mr 1,2,3, (4-83) (4-83) we obtain the equivalent of Eq. Surface Bottom /=/. (4-78). BDE = 2H cot cos d.z/K) It is mode is given by the factor sin first which is plotted in Fig. The factor sin {xJi/H) the depth of the source on the amplitudes of vertical-pressure distribution for each shows the influence of the different modes. The {Xr. and from the P2 \/c^/aî € = — 1 hVl -c'/al and using these definitions. f.<f<oo Second mode Fig. . and the disturbance at a distant point may be obtained by the superposition of waves arriving at the point along the oblique rays for which constructive inter- Taking the tangent of both sides of Eq. 4-6. From this point of ference occurs. (4-74) and (4-75) the factors 7(x„)/V^ give the relative excitation of each mode. each successively higher mode representing a higher order of interference. for all modes and for all frequencies. and infinite frequency in the first two modes. view it is seen that the normal modes are interference phenomena. 4-5. apparent that. 4-6 for the two modes. the free surface is a node in pressure and in horizontal particle velocity and an antinode in vertical particle velocity.A LAYERED HALF SPACE 141 results of Sec. In Eqs.<f<^ First mode /=/c f. Vertical-pressure variation in liquid layer for cutoff frequencj^/c intermediate frequency /. 3-1 From Fig. when h takes the form < z H. tion of an impulsive disturbance it is to represent the initial disturbance as the convenient to use the Fourier integral summation of a complete spec- FiG. . using the relations v^ == iv^. at the great distances for which our approxima- As would be modes and normal waves from the branch line integrals. this condition can be written in the form f^ 2^ cos ^ -f- X = 2mr (4-86) Equation (4-86) expresses the condition for constructive interference between waves traversing the paths shown in Fig. and the amplitude is found to decrease as r"". 4-7. and evaluation of (4-85) jdelds an amplitude which decreases as r~\ It Since sin' is interesting to note the physical interpretation sin d/lo of c the co/k condition cos ViH = 0. this result is invahd. With the substitution k = 2t and = = ai/sin 6. It was pointed out by Sato [145] and Officer [110] that under certain tions hold.142 ELASTIC WAVES IN LAYERED MEDIA expected. the steady-state amplitudes decrease as r"^ for the r"^ for the refraction residue. 4-7. which may be compared with the r~^ decrease for body waves in a homogeneous medium. (4-57). V2 = iv2. (4-54) the denominator of F{vi. However. obtain the write common denominator of the factor in brackets of (4-54) To we 1 vi 1 COS viH — < ibiV2 sin v-^H vx cos v^H + ibxV2 sin v^H (4-84) Thus. In the conditions the branch line integral can yield terms as important as the branch line integral £ in Eq. vV) is given by Eq. the term containing ViH is usually neglected in evaluating (4-85). the branch line integral in (4-54) ^['. to study the propagaGeneralization for a Pulse. when cos uiH = 0. As stated in Sec. [ F-'(fr) ^^ jjE 2 vt COS viH ^'!^yS'!^-^ + 5ip2 sm PiH r d^ (4-85) V2 is very small in the vicinity of the branch point. ^^^ = 2i5. Paths of multiple-reflected and refracted waves which may interfere. 2-6. ior example. for an arbitrary initial disturbance S(t). for (4-89) *iR = I [2 77 \ — y-. ^xn = < ^. x^h -j^ x„z sm -~ . which gives . we can make use of the Fourier transform g{o^) = .do: ^ (4-89) for To and a similar expression $2s for z > H. (4- In the problem of two liquid layers. ^.simple harmonic waves by a suitable choice of the initial phases and ampHtudes. whose steady-state solution is given by (4-74) and (4-75). r / e'""-""--^''' 2^ /-. we consider an arbitrary disturbance S{t). TT ^ C^'n) sm .. it is well known that distortion of the pulse will result from variation of phase velocity with frequency. do: _. instead of a harmonic function of time. H\/27rr /-— 2^ « J-a. The corresponding Fourier transform 9(od = • ^ /^ V-iTT / .A LAYERED HALF SPACE 143 trum of . o- is a parameter which depends on the energy of the explosive among is other factors (see Cole [19]). -jr tco) . At a suflici(!ntly great distance from the source the initial pulse will be transformed into a train of sinusoidal waves in which the frequency and amplitude vary gradually along the train. (4-74) would give. If the propagation is through a dispersive medium. application of the Fourier transform to Eq. z < represent the initial disturbance due to an explosion H we choose S{t) as follow^s: ^(0 = ' " ^ ^ ^ < ^ (4-90) where charge. (4-92) . If. Our derivation of the steady-state solution started with a spherical wave.R)]/R.y= ^ / S{t)e-"'' dt (4-87) to obtain the initial time variation in the form S(t) = -^^ j(o:)e-' j dc. which in an unlimited homogeneous medium could be expressed as 9^0 = exp [i(o)t — k. . / 9i(^)e ~--^ sm Vk„ -fj-n sm jj. (4-91) ((7 . It waves having angular frequency coo.^_^^^ •E(m) V{Xn) sin -~. where coo is the root of the equation To ^((^/ _ ^^r — 7r/4)/c?co = 0. given in a convenient form by Pekeris. (A-14) and (A-15) the value of (4-92) is obtained for large r: Hr „ ai - \ dU„/ai \ <r + icoo H f or h (4_93) I ^7 IJ < 2 < ^ Here y taken a:„ = k„ o}oH/2Tai. except for a narrow range of oo near coq. that is. (4-93). velocity is if the group considered as a function of the dimensionless parameter 7 For values Eq. U = is dco/dk.144 ELASTIC WA\'ES IN LAYERED MEDIA. similarly. By use of Eqs. leads to tives are negligible in î« _ - 4a 1 -sp cos \{a' [copt — K„r — tan~^ (uâ) — 7r/4] s^-^^Hr''' ^ + o:')Kj2-Kr%H\/2iv) dZJdy]''' . ^2r decrease with distance as r"\ an additional factor of r"' having been introduced to Eqs. is known as the group is related to the wave or phase velocity c by the equation U = At c + k"^ dr (4-94) large distances amplitudes can be calculated from Eq. because of the rapid alternations in sign of the exponential factor and the more gradual changes of the remaining factors in the integrand. at / t == r dKjdco. (4-74) and (4-75) as a result of dispersion. (4-93) is = - = c„<c„ ^ — = -^ (4-9o) of w near a maximum it or minimum value of the group velocity not valid. The exponent in Eq. the if d(UJai)/dy positive. In the neighborhood of a stationary value of group velocity. The amplitude of ir varies inversely It should be noted that ^m as the square root of the slope of the group-velocity curve. the next approximation. for given values of and the integral for ^^ will be zero. defining the velocity of a group of velocity. for involves the assumption that higher deriva- comparison with dU/dy. and. otherwise the negative sign is used. coo- upper sign in the exponential is and and are to be taken for The quantity U = dc^/dk = r/t.sin -^ . (4-92) will have a stationary value at a frequency ojq. obtain an approximate value for this expression we use Kelvin's method of stationary phase. r As shown by Kelvin (see Appendix A). ^>min.) . Function E{m) representing the envelope of the waves in the Airy phase.4 ' ""^ /£(m) 1.8 f<min.0 / / \ \ \ 0. 4-8.2 / 1. t<max. as a function of the dimensionless parameter y (4-95). f >max. The frequency equation (4-78) was used to calculate the ratio c/a^. name Airy phase to the waves associated with a The factor E{m) represents the envelope waves in the Airy phase and is plotted in Fig.2 ^ 1. Fig.8 0.6 v \ 1. from which C//a. 1. 4-8 from the data given by Pekeris.A LAYERED HALF SPACE for 145 < 2 < //. It is to be noted that the amplitude in the Airy phase depends on r~^^^ in contrast to r~^ for other waves in the train. {After Pekeris.6 0. Thus the Airy phase becomes relatively stronger with increasing distance. where 2 j2 Z„ = 27rf/: (4-97) <i7 ^(m) = m*[^-j(w) + Jj(m)] if < jy— ^ max t < jj— ^ mil > U„ (4-98j £'(m) = w*[/_j(m) — /j(w)] if i < f/n or < (4-99j m Pekeris has given the of the 4 i/ rfZ„ r /tti oil H\V~ JL (4-lOOj stationary value of group velocity.8 0.6 0.4 \ \ 0. ^^p1 '^^1f \ > /^T4 \ ri.2 1.ouunu 1.6 nc.8 V{x) c_ 1.1. density of bottom = 2 in Z= Wavelength water /= Frequency 0.) .7 0.5 -•2 3 0.L 146 1.1 ^Îli 1 s ^ 1 1 "^^^^ ^""^^^^fc 5 -^=1 ."1 J:3=^ _ ' — n| 2— - 1.6 1 1 1 1 V{x) = Excitation amplitude of first mod e lilt 1.3.5 if 1. 2. 4-9.5-L 1.0 1 u 1 1 . po/pi = 2. First-mode phase.8 < Vny J ::. az/ai = 3. 1.4 H «i 1 ii c 1.8 Eii^ 1^^^ / ''li -^^ /V 1 il 1 1 1 M 1 r= Phase velocitv of first 0. 1.(JJ/ai) velocity curves and excitation amplitude V{x) for liquid lajêr over liquid substratum. 1.9 0.5.0 "1 ] = 0'^ 1 .5 10 50 100 1-Hf/a. 3 ^ f \ \ ^^s 1 "1 1.0.05 as a function of parameter y.9 1 1 1 0.1 veiouuy 111 uuLiurri.0.(c/ai) and group. Fig.4 T 4\ "1" r 1 £ / V «2 V{x) U= Group velocity of first Gj = Sound velocity in water for _ mode mode aj.0.7 0. 1.6 // 1.0 ELASTIC WAVES IN LAYERED MEDIA 2. {After Pekeris. 0.i 0.3 ^^.b 1 j^^^s:^ ^^ '^ ^ -^ i^^m ^ "' II 1 IJSSS ^ 1.2 "2 . 4 \ \ Vi xi )^ \ - Vv^r ^ V 'X /\ '-T"— " == = r j=- ^ .0 . a family of phase.5 ^(^3) 1. Generally speaking. the greater the ratio ttz/ai.and group.0 is presented.) and group.9 s. 4-9. It is seen that the phaseand group-velocity curves cut off at a low-fnHjuency limit where U = c = ocz and that they approach is «.(c/ai) and excitation amplitudes for p^/pi = 2. a2/ai modes.6 \ 1..3 Y 1^ / iV \' \ 0.s for the first mode for various values of a.1 10 y=Hf/a. A striking feature the occurrence of a minimum on each group-velocity curve. sum In preceding sections the steady-state solutions were expressed bj^ a of residues of the integrand and a branch line integral corresponding to the branch point c = aa. 4-10. 0. Fig.^/ai when pz/pi = 2.:^P= '3 (6 u^ :^ 0. 4-10.]ocity curves for the three modes and for the ratio az/a^ = 1. asymptotically for large.The residues lead to the normal-mode solutions which predominate at large distances because of the factor /•"\ The normal- . In i'ig. To illustrate features of the higher first modes. Phase. 0.1 ^^ ^ t/.A LAYKKKD HALF SPACE 147 is obtained by the aid of E(l (4-94j.0 0. y.(U/ai) velocity curves = 1.7 0.- 1. given by Jêkeris.5 1 %/ u. a- w / / ^ r ^^r^.2 S» 1. Liquid layer over liquid substratum.2 1 \ \ \ \ s V \ \ \ \ \ ^.. number of modes is possible.and group. the minimum value being lower and occurring at a lower frequency.5 are shown in 1."^ ^ V- i^ ô - 0. {After Pekeris.8 y 1.-.5 in the first three Fig.1 s A> /^ ^.velocity curv(.8 17 0. phase. \ \ 0.4 1.". higher and for any given frequency only a finite modes involve higher frequencies.v(.0. —1 1 <o 01 ^ -* 6 -J fe tc 148 ../ \ . s f^f- \ '1 V •" N b ^ ^ ^ a.(Simn XjBJiiqjG) apn}!|daiB Ô'Ô o w-» IT) ft in •-( d o o d 1— wave 1 case . ^_ ^ *"" >> 03 O U O r^ T) :=! cr T) J . Time when = II 1 u ts / / * o" 1 ^ .^ A u 11'l / of 1 — Tim 'c^ / \ \ ^ \ water of branch (rider) on 1 / /\ • \ \ in Time beached 1 \ = y ^/ f \ J \ \ \ Wavelength T'o A ^'/ f \ \ I = are frequency frequency 1 1 . >^ of s t N ^ ^ wave both / of bottom 1 1 1 -- 1- • - -^ }S I t f / n ^/ A ' i .-"if)U T3 • 7- -P f\ n 1 - water for ">. o 1 of amplitude amplitude and / / \ ' arrival ^V Depth charge 1 1 J .i I 1 of H= Pressure Pressure -3^ / / ' \ T3 CO o c \ \^ ^ f1 1 y^.1 ^ > t^ -N \ k S beginning wave 1 the \ \ \ 1 — branch / f ^7 . o \^ o \ o o = \ ' 0) / wavelength low high hydrophone / / / * ^y -^.. </ A / t 1 1 1 water 1 in of in 1 1 1 1 \ \ ^ "c \ 1 Si •--' 1 -"I \ \\ \ 6 1 ^ Sl. time greater than r/a^ (that is. 4-11 as a function of travel time. bottom sound velocity = 1. superposed on the earlier low-frequency oscillations. Bottom sediment thickness = 220 ft. The results of Pekeris for two liquid layers have often been found applicable to studies on explosion sound transmission in shallow water. immediately after the arrival of compressional waves through thf.13 times velocity in water. This situation results generfrom the actions of filters in the recording system. water depth = 90 ft. high-frequency waves associated with the right-hand end of the group-velocity curve arrive. of the An experimental observation of Ewing layer obvious result of the Pekeris theory explains the early [35] that any accurate determination sound velocity for horizontal transmission through the surface frequencies.) . at which time the frequencies of the two super- imposed wave trains become equal. usually the fundamental. where the multiple refractions interfere constructively and decrease as r~\ All modes theoretically contribute to the motion at any point but in many ally cases recorded wave trains consist almost entirely of contributions from a single mode.1 sec V — ^'"-'•^-'•^^^-^\^\A/Ny\AA/V\A>^XAA/VVVv^^ Ground water Wave water Airy phase Fig. can be obtained from Fig. The wave amplitude and freciuency. At a. the normal-mode contributions begin gradually. (Courtesy of C. The branch line integrals contribute waves traversing the refraction paths shown in Fig. Waves (through two different filters) from an explosive charge of 55 lb observed at a range of 1. lower layer). The general similarity of the observed wave train wdth that must be made at the highest —-' ^•^'"^''^'^''^/^^n/^'vv\AAAA41aUV]^ 0. These waves correspond to the velocity curve (Fig.A LAYERED HALF SPACE 149 mode solutions vanish at c = U = a^. 4-12. forming an Airy phase. This is not surprising since core samples consist of unconsolidated sediments almost everywhere. An example taken from shots recorded at a distance of 17 miles in a water depth of 90 ft is shown in Fig. implying absence of rigidity of ocean-bottom materials. 4-12. r/ai. Source and receiver in water. The waves decrease with distance as r~^ except near the cutoff frequency. It is seen that the first of the the cutoff fre(iuency. Their frequency At the time rapidly decreases until a time corresponding to travel at the minimum value of group velocity. L. for example. 3-18. and because of the steepness of the group- velocity curve their frequency increases only very gradually with time. normal-mode waves arrive at / = r/aa with left end of the group4-9). increasing in amplitude with increasing time.030 times water depth. Drake. corresponding to travel in the upper layer. 8JnSS8Jd 150 . These expressions satisfied the conditions of continuity in (p and of discontinuity in displacement. we can introduce the potential (4-16) representing the source. To the boundary [118]. and thickness. conditions (4-19) and (4-20) we add the two conditions SitZ dz dz P2<P2 = P3<^3 = Hi-\-H2 (4-101) The former layer thickness H is now denoted by shall H^. 4-13). <p[ such as (4-36) and (4-37). Three-layered Liquid Half Space. by expressions containing trigonometric functions of the first layer. 4-14. t h 'S X Hi ^ Pv «1 P2< <^2 1 P2< »3 > 'z Fig. [116] and Press and Ewing the preceding section can be applied to the Figure 4-14 shows the notations The method developed in new case with slight modificawhich will be used. Point source in three-layered liquid half space. for a layer of finite the potential was alternatively represented z. and (4-96j striking (Fig. lagoon bottom on Bikini He found that when the thf. a low-velocity bottom. Cases where the simple liquid-layer theory has been found inadequate have been attributed to a layered bottom. respectively. (4-93. = h (4-102) Jo . [26] used the observed dispersion to deduce properties of the atoll.ory was appli- could give useful information on the bottom to a depth comparable to water depth. 4-3. we have five arbitrary functions Q in expressions (4-11) and (4-13). that is. In order to satisfy the five boundary conditions. we made use of two different and (p[' for the potential above and below the source. we put [see (4-39) and (4-40)] dcpi^ «Pi <Pi dz dz 2 [ Jo(kr)k dk at 2. In the preceding section. In expressions where the source is located. or a solid bottom. This problem was investigated by Pekoris tions.A LAYERED HALF SPACE 151 is computed Dobrin cable it for the first mode alone from Eqs. F.152 ELASTIC WAVES IN LAYERED MEDIA choose <pi We surface is satisfied.Q\ sin V2{Hi -\- E+ iv. • • • is reduced to To determine these factors we substitute (4-103) into the remaining boundary conditions (4-19) and (4-101).lOo) A A Hence sinpih cos Vih = BsinVih + C cos u. <z <H. D.e~'''^"'^"''F = . C.+H^ <^3 = f Jo F(k)e~''''' dk for H.H^ <z > where j5.- = -iVk'' - k~~ if for A- k^ (4-104) The conditions (4-102) are satisfied by (4. of unknown factors A. E. B. (P2 (4-103) = Jo D(k) J o(kr} sin V2Z dk -\- I E(k)Jo(kr) cosV2zdk for Jo H.h — B cos Vih + C sin Vih = 2k — (4-105) A ~ B = (7 — cos Vih (4-106) sin vji = —- and the number four. ^ Ps = h (4-107) we 5i obtain four equations sin viHx B — B — -\- sin V2H1 D — D cos VoH^ E = 25 Vi — sin Vih 7c cos V^H Vi COS ViHi sin V2(Hi V2 cos V2H1 -{- V2 sin V2H1 52 H2) V2 cos?2(//i -\- H2) D D - -{ 82 cos V2(Hi V2 E = 2k sin Vih sin ViHi + H2) E . Putting ^ P2 = 8. such that the condition of vanishing pressure at the free The solutions may now be written viZ (p{ = Jo I A(k)Jo(kr) sin dk ior < z < h <pi' = Jo I B(k) J o(kr) s'm ViZ dk + I C(k) J o(kr) cos V^z dk Jo iovh<z<H.-\.e--'<^'+^=>i^ = H2) /^_ip. ^3 = 25i52 [ •^° Joikr) ^^^ [S cos ^2/^2 - sin u. we obtain the branch line integral in the form JS [ = ' f Jo H'o'\^r)N^ d^ + [' Jkat m'\^r)N^ d^ (4-117) where . sin v.H2y'''-"'-''''k dk (4-114) The integrands in (4-111) to (4-114) are even functions of discuss one of these solutions. and therefore the corresponding branch Une integrals will vanish.V = -^^ [P25.g. h) -\- 8. - z)]\ (4-118) . - h)]k dk (4-111) ' <p'/ = 21 ''' Joikr) "' [SVi cosPi(i/i ^ 2) + — 5iF2 sin ?. ^[' = f - 2Ti ^ Res (4-115) The residues are determined is. + — J'l'S' cos ?iff 1 (4-110) solutions = 2 [ ''' Joikr)^-^^ "'^ [Sv. the notations • • 153 . which by (4-110). We now Since there e. coHp.(/Z'i — z)]k dk (4-112) [»S <P2 = 25i / Jo{kr) —^^ cos 1/2(2 — Hi) sinVziz — Hi)]k dk (4-113) ''' . by the roots of the frequency equation. and. As before. in more detail.Hmv. by similar transformations used in Sec. as in (4-54). we can write. - z) +v.(H. F into the expressions (4-103) and using 02V3 + 11^2 tan V2H2 and V = we obtain the <pi 5ii?2 sin F. (4-112).A LAYERED HALF SPACE Substituting the factors A.iH. point kaz = one branch line integral which corresponds to the branch ^/dz. we replace the Bessel function Joikr) by the Hankel function HQ^\kr). is î and P2.v. 4-2.(H..//. F = Again use has to be made or t&n PiHi = -P1S/V2 (4-116) of real roots only.S cos v.iH. Then by Eqs. Comparison with Fig. we find on applying we used in Sec. a c = a3 ov k = ka3. The case for which the intermediate layer has a lower sound velocity than the first layer was investigated by Press and Ewing [118]. that is. 4-9 shows that the group-velocity curve could be approximated by treating these two layers as a unit having the same composition as the upper layer. they determined the phase. (4-104)]. it has a low-frequency cutoff at t/ = c = ^3 and a minimum group velocity. Considering the possibility that a low-velocity sea bottom could account for this phenomenon.1. (4-116) and (4-109) tan v. shown for the where H2/H1 = 0. 4-15. where group-velocity curves for a three-layered half space are three cases indicated. a^/ai = 3. Only the low-frequency branch of this curve was observed experimentally. In case (a).154 ELASTIC WAVES IN LAYERED MEDIA to the As the same rules « iui wave trains represented by residue terms. and it approaches eta asymptotically at high frequencies. Since by (4-116) the phase velocity c is a function of frequency or of k. >r^ N7 sin vji v^S co s viJHi — z) + V2^x sin v^{H^ — z) (dV/dk)„ (4-119) The frequency equation (4-116) takes different forms for c > 0:2 or c < aaNow. Some results of Pekeris' calculations are presented in Fig. aa/cti = 1. where H2/H1 = 10.H^ tan VoH^ = -^ O1V2 (4-120) from which the cutoff frequencies can be computed.1. where the layers are of comparable thickness. They were interested in the fact that in some areas the "water wave" consisted of a brief burst of high-frequency sound which did not show the dispersion and Airy phase normally found. the complete three-layer calculation must be used except for very high modes where any problem may be approximated by considering two layers at a time. the second layer is thin. Pekeris has showTi that under certain conditions the group-velocity curve now has two minima. In cases such as (b). absorption of high-frequency sound or possibly sublayering . and its properties are nearly the same as those of the surface layer.0.and group-velocity curves for a three-layered Hquid half space with aj < oci < 0L3 and H2 = 5Hi. we shall have v^ = [see Eq. In the first mode there is found a curve not unlike that which would be obtained if Hi = 0. we may approximate the high-frequency end of the group-velocity curve by considering H2 = ^ and the low-frequency end by considering Hi = 0. 4-2 that they are given by the sum 7-7(2) /. the group velocity of each of the normal modes for a three-layered Hquid half space may be computed. For case (c). 5 1 10 Fig. 0. 4-15.01 0. {After Pekeris.0 10 % 'x a ^ — i \ .10 3.A LAYIiKEU HALF SPACE 155 1 1 1 1 1 1 1 1 r r r r 1 Number Caoc of «? "l "3 «! Hy H. U /ai.1 1.10 3.0 I.) first mode in a . that is.1 0.IC ) 3.0 layers a b c 3 3 1.05 0.0 3 1. Group-velocity curve.y ^' N \ ^ / ^X ^ „-s-^ o* ^/ 0. as a function of y for the three-layered half space. Pi. provided that the wave length was great compared with the water depth. As in all other problems treated we assume that the boundaries are parallel planes and medium which composed of a liquid layer and of an underlying solid half space. P3. but he also neglected the shorter waves which are prominent on many seismograms. He found that the frequency equation expressed the condition for constructive interference between the primary wave Pi in Fig. 4-16 and the sum of the multiply reflected and transmitted waves P2. which traveled at velocity the authors proposed that reflections at H1H2 interface at grazing incidence could explain the retention of high-frequency energy in the water layer. Officer [109] has derived the frequency equation three-liquid half space by the use of rays and solutions for the and plane-wave reflection and transmission coefficients. Scholte [152]. 4-16. In a note added to that paper. while attempting to explain microseism generation by transfer of waves. • • • Fig. using the theory of plane waves.156 ELASTIC WAVES IN LAYERED MEDIA in the aj layer being suggested as an explanation for the absence of the the properties of the direct waves. Jeffreys proved from Stoneley's equation that there exists a minimum of group velocity at some period shorter than those investigated by Stoneley. This problem applies directly to the propagation of earthquake surface waves across the ocean and to the transmission of explosion sound in shallowwater areas when the bottom is solid. the To deduce aj. Sezawa [162] obtained an approximate solution for the propagation of cylindrical waves. calculated phase velocities for a water layer assumed to be 3 km thick over a group and solid substratum. in a plane normal to a wave front. He confined his attention to the longer-period Rayleigh waves and concluded that the effect of the water layer was unimportant. 4-4. in this chapter. high-frequency branch. To investigate the effect of the ocean on transmission of Rayleigh consider the problem of propagation of a disturbance in a is [193]. Ray interpretation of period equation for three-layered half space. Calculations will be given for a source of compressional waves located in either the liquid or the solid. Stoneley . Liquid Layer on a Solid Bottom. presented curves of phase and group velocity for the first and second on a solid bottom. 4-2 and Fig.H+d) Jz Fig.52. Longuet-Higgins [90] applied a similar theory to the horizontal transmission of microseismic energy across the oceans. 43] used this theory to explain some features of the propagation of Rayleigh waves across ocean areas. normal modes for plane waves in a liquid layer superposed For application to transmission of explosion sound in shallow water they gave calculations for an impulsive source within the Hciuid layer. Pi 2=H0-2. The displacements are expressed in terms of the potentials <pi. 4-17. Following Press and Ewing [120] we use the notations of Sec. Thus dtpi dr Wi 3^. in a study of microseisms. ^2. Not being concerned with details of horizontal transmission. he gave no mention of dispersion or group velocity. Introduce 2=0- «i. Compressional-wave Source in the Solid Substratum. the two velocities a^ and 0^ for the propagation of waves in the solid layer. considered the combined effects of gravity and compressibility in a layer of water in contact with an elastic-solid bottom. they extended the theory in order to include the case of an impulsive point source of compressional waves located within the sohd bottom [120]. P2 S{0. 4-17.A LAYERED HALF SPACE energy from gravity surface waves to elastic 157 in the waves bottom. Ewing and Press [39. Compressional-wave source in solid substratum. in a search for Airy phases from submarine earthquakes. and i/'2. Press and Ewing [117]. dz (4-121) dip-2 W2 dr dz dr dz OZ . Later. 2 "T ^Ms ~r 3 I %2 7..2 ^^^2 ^ (4-133) Equations (4-125) and (3-106) and the wave equation have been used in .z ior < z < H (4-126) where Vl = kal k' (4-127) A time factor exp (iwt) is understood.)r = (v^X atz H H H in Sec. 4-2. (4-123) (4-124) (4-125) In order to satisfy condition (4-122). to -\- With a shght change of notation we can use the expression (4-16) represent spherical waves emitted by a point source at r = 0.= f Jo ^1^2 ê-'''''-''Jo{kr)dk-\. as dk <Pi =- [ Jo A{k)Jo(. = = = = (4-122) = = W2 atz atz (pj^ (p.[ IV2 QMJo(kr)e dk (4-128) Jo = Jo f S2{k)Jo{kr)e-'''''' dk 7 where If 2 -2 V2 _ — "'a2 J. Thus.kr) sin v.158 ELASTIC WAVES IN LAYERED MEDIA with the boundary conditions (^1=0 Wi at 2.2 at _ — X2 d^<P2 n ^^2 ^ 2 d I \p2 -^ a2 ot 2 r.. <P. we can put. z = H d = h. we have k 4 'Joikr) <P2 Jo [ dk IV2 + Jo [ Q^Mkry dk (4-130) ^2= The boundary f Jo S2Jo{kr)e '"'' dk conditions (4-123) to (4-125) at z cosiJiiy = H now take iv2Q2e"'"' the form (4-131) (4-132) ViA = ke"""''' + k^S2e~"'^ " — = I 2'a)2Q2e~''"-" + {.2 2 V-y •C — n /Cflo ~~ K (4-129) < i/ + d. H {2k^ . vx and (4 134j is cos vxH iv2 —]z v'^ A = Piw sin j-ji/ 2/x2^" 2iv^ - fc' (4-135) ~ P2'j^ —2fjL2k 11)2 and the values of A. we can write for (4-135) 4 _ A(k) PlCO lV2 = ^^^^sinv.V2 + — (2^ - k^Y] cosv. Using M2 = P2182.// + {îJ-'ik'' — pîiJ'^Q.if'^^ — The determinant l\i.3'i/ (4-139) Now Q2 write (4-138) in the form = ^T" e %V2 — 2 ^— -^^ e P2/32 cos PiH (4-140) IVylS and substitute in (4-128) to obtain (P2 = Jo f -e-''^''-"-''J.{kr)dk IV-l (4-141) e e Jo [ ivi ^ k Vrpo^l cosv. R" = r -[- {z . After a rearrangement of terms.A LAYERED HALF SPACE deriving E(i.H + df. (4-132).iK ti'2*J2e = 9 t.H (4-136) P2 and the coefficients A.) loO takes the form Pi(j)Â sin P. S2 are given by the expressions A Q2 2p20) {2k kffy)k _jT (4-137) = -:—:& tI'2A \ I o2 /32 sm t/i/f (4-138) + -y j'iP2i82[4/cî'2J'2 — (2fc^ — /câ)^] cosViH 4^(2^X2^^ — P2<J^^)V\ cos ^liJ -.H + i7. .{kr)dk+ lV2 k Jo f -e-''''-"*''J. (4-l'4?.k%y Joikr) dk The second term may be image where interpreted as a spherical wave emitted by the is of the source in the interface.p2^2[4A.2 ^ 2 (4-134) ke tj'2 of the three e(iuations (4 131). Q2. Q2. (4-133). Its simple form exp ( — ika2R')/R'. S2 in terms of k and other parameters can be found if A F^ 0.73d + iT. - kI (4-150) . and (4-128) with (4-137) to (4-139) we can find the displacements q and w.^ dk (4-146) before. The residues correspond to the poles = k„ given by the roots of the equation T{k) = (4-147) As we have seen.H e-''''Joikr)k iV. = 2 rl Jl X \l 1 W^(K„)e-^'"V^"'-^-'^^^ for each (4-149) r „ V/c„ V2n where the phase velocity c„ c„ and the factor vl mode are given by = - = kl. branch line integrals and residues. given by (4-142) and the third integral in (4-141). the amplitudes of r~^.(-H) cos v. the terms corresponding to branch points. (4-141) may be combined as follows: Jo ^ cos IV2 - H)e-""'Jo{kr) dk ior H<z<H+ + d<z < d (4-142) or 2 r ^ g-. (4-126) and (4-128).{z in Eq.d 2ip'2 J. (4-126).160 ELASTIC WAVES IN LAYERED MEDLA first The two terms v. waves determined by branch line integrals diminish as Since we are interested in an approximation which holds for large values of r. on putting A(k) = T(k)p2P^^ eosv. (4-121). the integrals can be represented by the sum of A. where the expressions (4-137) and (4-139) have to be inserted for A and S^By Eqs. and \p2 by (4-128) to obtain. and asymptotic values of displacements are obtained as follows (the time factor 5h is written again): (4-148) = 772 \\— 2^ ^~i= QMe e ^.{kr) dk (4-145) Wh = -2k}2 As 2k' - k '' I \^j.-. Thus for the solid bottom z = H we make use of (P2. w^hich by analogy with earlier results represent waves mth phase velocities az and jSz.H (4-144) two expressions Qh for the displacement at the interface: •Pi / — — 2/c^2 k02 Vi Jo T(k) LP2 ^ tan v. Then only the residues are computed by the methods used earher.d Jo(kr) dk Jo lV2 iovH cx> (4-143) The functions <Px and yp2 are given by Eqs. are left out of consideration. - e< 7iT (4-155) k^ tan"' K < H Vk'l. Each of the roots /c„ of (4-154) can be expressed in terms of oj or T.k^ < nir . _ :2.ip.cVttz) . 1 Vc'/a? sec 2 - «„//(! C„/q:i .tan"' K > HVk\. .A LAYERED HALF SPACE 161 and QW kJI c„ P.k^ l)Tr < (4-156) the number of roots reduces to 2n. with the elastic constants of the system as parameters.^t êja\ - tan KnII 2 2^ 1 (4-151) tt2 2 1 /^(/Cn) /32 --i «2 1 (4-152) with Pi c^ 1 /^(O + 1 P2 182 _ c„/o. ci Vl - cl/al 1 I ic: R{K„)^f. there are 2(n + 1) real roots on the axis symmetrically distributed mth respect to the origin. In the case (n - n-K H Vkli . As mentioned earlier.1/ tan kJI Vcl/a.clA«2 Vl . ^2 . A.c„7/32 1 -22 The period equation (4-147) can 1 i _ ^ 2 (4-153) "2.dla^ . If —K (n riTT - l)7r < HVkl. 1 . (4-154) It defines as usual a relationship between the period T = 27r/c„K„ = 27r/aj and the phase velocity.1 «2 IT KM\r^ \Cn VI-'] . . Schermann [151] proved that there is a finite is the right-hand member of (4-154) and if number of real roots.T . now be written in the dimensionless form tan [KnH\H^ — 1 Cn PiC^Vl — 4a cl/al _ 1 2 1 „2 12 "2 ^. for two modes when Pi/pi = 2. 0:2 = v3/32.1 0.8 \. respectively.0 N 1.6 0. will now be discussed studies of water-covered in detail. and V 3/32.4 \ \ \ \ \ \ r \ First ^1 \ ^ \ 1. \\ >^ Sp cond mo de 0. Phasefirst and group-velocity curves 2. 4-18. 4-18 and 4-19 are shown the results of computa2. (4-94) were used to obtain the phase velocity c and the group velocity U in terms of kH.0 0. a2 > commonly encountered in geophysical ^2 ^ c > ai.8 \ ^y / >^ . The phase and group velocities of the first mode (n = 1) will approach the velocity Cr of Rayleigh waves in the solid layer as 7 —> or as the . S s \. ai/02 = \/3.2 \^ Tr "' \ u "1 \ ^ \ \ moc Q ^ 1 1. In Figs. \ 1.162 ELASTIC WAVES IN LAYERED MEDIA case A areas.i. /32 = 3a:i.6 '^ ^N \ \ \ ^ \ 1. The period equation (4-154) and Eq. Following Pekeris. xVy 1 <.5. 132/ai = tion for the numerical values: P2/P1 = 2.0.5 10 Fig. a2 = conditions for a granitic and basaltic ocean bottom. ^2 = 2a. which represent the approximate P2/P1 = 3. Liquid layer over solid substratum.5. we express c„ and k„ in terms of a dimensionless parameter 7 = CnK„H/2irai. /32/ai = sented by dots to the right of c/ai = 1 on the first mode). 4-19.6 \ \ \ \ 1 1 / / C \ \ 2. the c/ai and U/ai curves extend < 7 < co and that for large 7 these ratios approach the over the range Stoneley-wave velocity O. (4-127) is imaginary. 3-3 and Fig. the amplitude in the first layer now decreases with distance above the interface. U/ 0.1 0. in contrast with the results for liquid layers. F.2 0. a2//32 = a/S. 4-18 where these waves are repre- 3.. curvês for two modes when P2/P1 = 3.5 10 Fig. Liquid layer over solid substratum. Pôr c < a..0 nod ^-.6 V/ I < 5ecorid rr lod e .A LAYERED HALF SPACE 163 first layer.. and the waves may appropriately be called Stoneley waves (see Sec. Phasefirst and group-velocitj' 3.^ 0. Note that.2 \ l\ 1. Coulomb [20] and Biot [6] discussed the . the waves corresponding to this mode are termed first-mode generalized Raykiigh waves.QQSaj. \ '^^f-irst S.0. in Eq.0 Ns V \ \ i:::.4 j_r ai ^ \ c \ '^ a. 1.^^ 2.8 \ \ Ol \\ \\ I ^ J 1. wavelength becomes very long in comparison with the thickness of the As long as c > a. velocities are further characterized values. (4-157) ^. (4-160) A J] n r J-cc -J-.W(K„)e-''"'V"''-'"'-^''' \/k„ dco (4-161) Approximations for these integrals can be obtained by using Kelvin's method of stationary phase. in the form 9h = 4^ \ - E E „ [ 4= Q(0^(co)e-^-V^"'-'"'^^^^^' -4= \/k. The solution for an arbitrary initial disturbance can now be written. The group in Fig. In the second-mode generalized Rayleigh waves (Fig.= W^ \/^ H \7rr [" j_co W(K:)g(o. In general. 3. For longer periods. Then Eqs. we shall put g(o)) = A. and attenuation occurs with increasing distance. as discussed in Appendix A. 4.164 ELASTIC WAVES IN LAYERED MEDIA theory of propagation of Stoneley waves along the sea floor in greater detail.^ g(co) do. We find for d(Un/(Xi)/dy qn > (4-162) = = -^ Z ^^ S 11 r „ L„Q(Oe~''^"'sm (coof - kj-) Wjj L^W{K„)e-'''"' cos (coo^ - K^r) (4-163) . increasingly shorter cutoff periods. As c and U approach ai. corresponding to radiation of energy into the bottom. • • = sively shorter. as shown 4-19 for the first by stationary and second modes. (4-157) and (4-158) take the form Qh = JT2 \— 2 1^ / —^ Qiicje e do. 4-18) for a value of c = U = ^2 is 7 corresponding to a cutoff period. (4-158) where the Fourier transform given again by of the initial time variation S{t) is g(w) = -4= V27r •^-- f S{t)e-"'' dt (4-159) Assuming that the impulse developed in an earthquake can be represented by the condition that S(t) vanishes for all but infinitesimal values of / in such a manner that /-co S{t) dt = A. the periods There can exist higher modes of propagation having the same cutoff velocity c = U = ^2 but each (n •). as in previous cases. the periods of the higher modes corresponding to a given phase or group velocity become progres- become infinitely small.)e-''-V'"'—''' do. Only the final results are now given. k complex. I y - VmI Z = yu — «! dU 2ttU' dy _ (4-169) ^ r ^M Tj ~ 1 (4-170) The second velocity.^ and H' fdZ L2x \\dy (4-171) These expressions have been used to compute the amplitudes of the component displacements at the bottom for an impulse produced by a . and These expressions hold Tolstoy [119] are qH = q2/3^2^5/6 £ L^Q(K„)E(m)e 2 + — "'""^ cos iojot — Kjr + ^ 1 (4-166) Wh = where 02/31^2 -576 LiW(Kn)E{ni)e''"''"'^ cos (oôt -«:„/•- ^ (4-167) E(m) = = m}[J-^(m) Ji(m)] for t > < or U„ or Ur.)/dy < 4q!|A Qh = -^/- S KQU:)e-''""' cos /^„Tf We"''^"' sin (cooi - /c„r) (4-164) ^" = %^ S (coo^ K„r) (4-165) for a large r and for t sufficiently removed from a time corresponding to propagation with a stationary value of group velocity. The final expressions for this phase as given by Press. The to be evaluated at U^^^ or U^. Ewing. The train of waves corresponding to this group velocity is called the Airy phase.A LAYERED HALF SPACE 165 where L„ = 7«i C„ £ii_ djUJaM ] -i dy \l Ul and for d{UJoc. expression in (4-169) vanishes at a stationary value of group of is But only the derivatives subscript M denotes that a function u = Kn Z are involved in the results. ^ min > < t (4-168) t E{m) with m'[/_i(m) /i(m)] for t m= 4A/7r 3\/|t^Z/rf7| ^ to. Respectively.01 ^ y^ yy ^ ^-J^ = H ^*»^^ — ~. and at distances of 1. 4-20. and 4-21 were given for po/pi = 2. The only difference will be Suboceanic Rayleigh Waves: First Mode. 182 = 3ai. a sound source in the liquid layer in the amplitude functions.166 ELASTIC WAVES IN LAYERED MEDIA point source of compressional waves located at a depth h = H or h H = 0.05 0.000 km and 10.5 7 Fig. P2 = 2.000 km. 0:2 = V3/32. 0:2 = v 3/32.i A^~2 10~' for first mode as a function of 7 at a range of 1.5.1 0.000 km. is treated. Figures 4-20 = 5 km. ^2 = 3q:i. section can be The theory developed in this appUed to earthquake Rayleigh waves propagated along a .5 pi. 0.H 14 12 10 i / / ' h= 0- 7 / t \ ^/ ' t y 0. Discussion of the features of seismograms which correspond to various parts of these curves will be deferred to the section where the case of "lOz^. Vertical displacement w^^h in units 4q. Liquid layer over solid substratum. . in the Fig. 1953. 1 \ s- I h= // IJ 0.5 Fig. Liquid layer over solid substratum. 17.A LAYERED HALF SPACE 1G7 path that is largely oceanic. 2." which [e.g. 100)].500 km in the Pacific Ocean and about 4. The seismograph has three matched 35 30 25 c\ 20 1 \ ] 15 \ 'A=0 \ 10 / f\ 1 \ \ .000 km across North America. The study of these seismograms shows that: 1. We reproduce first in of Aug. mode as period T^ components each having pendulum period Tq = 15 sec and galvanometer = 90 sec. has long been considered an unsolved problem Ref. The azimuth at Palisades is about 260°.1 H 10 Â 0. 4-20). 4-21. 9!) Jeffreys (Chap. pp. It resolves the prol^lem of the "coda. The path covers about 8.000 km (see Fig. Tonga 4-22 the Palisades records from the earthquake Islands.. The orbital motion is proper for Rayleigh waves arriving from the west. Vertical displacement for second a function of 7 at a range of 1. 12. 168 . Since dispersion can modify a pulse into a train of sinusoidal waves whose period varies gradually along the train. For all other stations. From these data. and having a substantial portion across ocean basins the Rayleigh-wave trains could be com. layering. .56 km/sec. It will be discussed in more detail in Sec. can produce disis persion in Rayleigh waves but the difficulty that the lowest possible group velocity is only slightly less than the lowest shear-wave velocity in any layer. This continental travel time was computed b}. Guided by experience on dispersion in explosion sound transmitted in shallow water. 4-5.8°S 7 155. For the direct paths to Honolulu and Berkeley. 41] reopened the question of the effect of the ocean water on propagation of Rayleigh waves across ocean basins. 4-23). The period of the wave varies gradually within the train. Solomon Islands earthquake of July 29.7 km. and Palisades.use of a dispersion curve derived by Wilson and Baykal (2 10] and refined by Brilliant and Ewing [11]. epicenter and magnitude Seismograms were studied from Honolulu. the period decreasing from about 25 to about 10 sec. In all cases the seismograms are generally similar to those reproduced in Fig. For Honolulu and Berkeley the path for the direct waves may be considered entirely oceanic. IIALP^ SPACE 169 The waves first are markedly sinusoidal and clearly exhibit dispersion. and the inverse waves (coming along the major arc) were registered at all stations except Honolulu. no rock layer was available with sufficiently low shear-wave velocity. group velocity has been plotted as a function of period in Fig. 4-22. 4-24 for dispersion over the oceanic and continental parts of the path.52 km/sec. 4-24. The excellent origin time 23H9'°08'. with the water depth H = 5.pletely accounted for by the curves of Fig. They found that for paths like those shown in Fig.A LAYERED 2. An example of this method of analysis is provided by their study of the surface waves of the 6.0. Seismological Association). However. (Jesuit depth 75 km±. a^ = 1. /3.95 km/sec. and pa/pi = 3. (4-154) and (4-94). 4-24 is derived from Eqs. it is natural to attempt to explain the coda as a dispersion phenomenon. the total travel time was corrected to represent the oceanic-path segment by subtraction of the time required for travel across the continental segment of the path. Berkeley. 1°E. As will be shown later in this section. Tucson. The part of this wave train in which the period is almost constant has been called the coda. Ewing and Press [39. 1950. 4-23. az = 7. group velocity across the ocean was calculated as the ratio of distance to travel time. The direct waves (those along the minor arc of the great circle) were observed at all stations. and arrival time can be determined as an empirical function of period. The theoretical dispersion curve in Fig. rapid at The decrease is very and so gradual at the end that the period could be judged constant. = 4. which lie near a single greatcircle path through the epicenter (Fig. such as that present in the crust. a " a 2 a 2 c O 170 . IO) > c 0) . Q- - O H - 1 1 r. / / / / / - CW 1 \ \ \ 1 c o '(/) — 1 1 > 3 O CO O O C/> a ra CO "to T3 (O </) _ CQ < < D + o m X QJ "ca Q.iC «— r^ in in —> V ^> :^ ^ - 1 1 1 1 1 1 1 1 CM in 00 '^ "* St o ^ <£) CO CM CO 00 OJ -^ CvJ o CM •09s/'uj>j '/4po|9A dnojg 171 . oj v.-• (U . O - 1 ^ r^ -a iC o O) \I Is^ Si-' 0) \+ o Q) x: 1— \ °^ 1 \ \ \ \ o ^<] \ <^ o O c - / ' \ v Vq ^^^ \ \ S5 X / _ . °Vx A> \)Hr^l? /" - in E m.-i II -^ CD •-< II « . "^ \ s ^ \ \ inu3 \ \ cTi in \ rv! -^t CO X \ II 11^ ^ o II ^ ô./c ^ E \ km..^. -*0 < xa _ II <M CM e CO.î: > .— .> *-• •— Qi ir ^ u _2 o Sr <L) T3 — 3 (- ^ OJ ? IS) o <" i- </5 VO) o c/) O) (U c: X c 3 o (= 5 o c 9 ^ n 1- "5 >. An objection has sometimes been raised to the idea that long trains of Rayleigh waves. Likewise it has been maintained that the absence of long-period surface waves at the smaller epicentral distances precludes the possibility that dispersion can explain the appearance of the train at great distances. Ewing. Their method consists in plotting the dispersion curve for each seismogram on a grid. 4-25.172 ELASTIC WAVES IN LAYERED MEDIA this theoretical curve agreement between and the observed oceanic dispersion for direct and inverse paths demonstrates the essential identity in crustal structure of the Indian.2 km to the southwest and consider the probable is error in their results less than 50 per cent. troughs. which must be used to obtain the best agreement between observation and theory.7 km. The portion of the coda in which the period seems to remain constant results from the steep portion of the dispersion curve where a large change in group velocity occurs for a very small change in period. and the underlying rocks are also represented by a single layer whose properties are very near to those of the ultrabasic rock bounded by the Mohorovicic discontinuity. The method of deducing arrival time as a function of period from the seismograms consists of numbering the peaks.9 km/sec (basalt) and a very thick underlying layer in which a^ = 8. which lies at 30. Oliver. it is From any seismogram showing possible to estimate the mean a good train of oceanic Rayleigh waves thickness of the suboceanic sedimentary and Press [111] applied method to the Honolulu seismograms for earthquakes in the circumpacific belt. The liquid layer. A puzzling feature of Rayleigh-wave propagation in the oceans the .95 km/sec represents a sort of average of an actual structure consisting of 5 km of a layer with velocity 6. such as that shown in Fig. and Pacific basins along this depth 5. They found mean sediment thicknesses ranging from 0. wave train and plotting these numbers The slope of this curve gives period as a function of from which the group velocity can be computed. This representation of oceanic crustal structure involves two simplificaThe sediment of the ocean floor is included in the liquid layer. and zeros of the against travel time. 4-6. as a measure of the mean floor thickness of the layer of unconsolidated sediments on the ocean tions. exceeds the mean depth of water in each basin by roughly 1 km. could occur solely through the effects of dispersion.1 km/sec. The view has often been expressed that a long succession of sinusoidal waves of almost constant period was due to some resonant phenomenon.to 40-km depth beneath the continents.4 km due north of Honolulu to nearly 1. along the selected great-circle path. Atlantic. it will be shown that the value cca = 7. such as those studied here. and subtracting layer along the propagation path. this the mean water depth from the liquid-layer thickness read from the grid. travel time. which may be taken great-circle path. In a later section. even if there are some special conditions which help these short-period waves to enter the water. Thus.56 km/sec. a depth of focus several times greater than the water depth would reduce very strongly the amplithat. 4-19.to 12-sec gap in the spectrum still remains to be explained.0 1 1 1 1 1 1 1 1 1 1 I I 1 1 1 1 10 15 20 25 30 35 r in seconds Fig. a train of waves with periods less than 1 sec which travels with the speed of sound in water (see Sec.52 km/sec.e. 182 = 4. "WTien continental Rayleigh waves. Great depth of focus has been suggested as a possible explanation of the absence of Rayleigh waves with periods shorter than about 12 sec. Group velocity of Rayleigh waves as a function of period for various depths of the liquid layer and for a2 = 7. absence of first-mode waves in the period range from about 1 to 12 sec.. the 1. 4-20 The important question is an ocean with a uniform depth of 5 km. reach the coast they do not continue out to sea as first-mode . minimum of group velocity in whether these waves are not generated by the source or are not transmitted across the ocean because of the effect of some factor not accounted for in the theory. is strongly by most submarine earthquakes. with large amplitudes in the period range 1 to 12 sec. H P2 = 2pi. 7-2). 4-25.A LAYEKIOIJ HALF SPACE 173 - 4.0 / // / l"^ // /s! /s! /v* -/ III 1 1 1 1 1. But it because the excited seems improbable that depth of focus is the correct explanation T phase.0 ^^^^ - ^3. This represents a gap in the spectrum corresponding to the part of the dispersion curve lying to the right of the Fig.95 km/sec. i. ai = 1.0 E - Li /o // Ao / /'^ // l^ /v /c? /<o /*^ /<o" /o Ao /to" /Co" R 2. and it may be seen from Fig. in tudes of these waves. Despite a fairly thorough search. Newfoundland. an alternative explanation was sought. Compressional-wave Source in the Liquid Layer. (4-122) to (4-125). The observed travel time fits the hypothesis that they are Love waves reflected from the Grand Banks off or detecting distant storms. 4-2). (4-121). They have the combined characteristics of Love waves and the second-mode Rayleigh waves and will be discussed in Sec. as will be seen later in this section. Ewing. and the boundary conditions are Eqs. Thus we write. 4-5. marked "water wave" and "Airy phase" (minimum group velocity) are prominent. 4-20 and 4-21 that the Airy phase should contribute to seismograms of oceanic Rayleigh waves a prominent wave train with a period of about 12 sec and a group velocity of about 0. . no waves which even roughly fit this description were found except on Bermuda seismograms of West Indian earthquakes (Press. as before. and Tolstoy [119]). In (see Fig.7 times the speed of sound in water. Suboceanic surface waves with periods 6 to 12 sec are observed on seismograms from earthquakes in certain areas. It may be inferred from Figs.z + [ Jo C(k) cos vrz]Jo{kr) dk foTh<z<H (4-173) <P2 = Q2ik)e'''''Jo{kr) dk (4-174) iorH ^1^2 <z (4-175) = [ Jo S2(k)e'''''''Joikr) dk Condition (4-122) at the free surface 2 = is satisfied by the assumed form of (4-172). Since these waves have not been observed elsewhere. This fact severely limits the usefulness of microseisms for tracking No gap in shallow Rayleigh waves water of propagation in occurs spectrum in the short-period waves between the points seismogram this 4-12). We shall now consider the second case mentioned above where the point source is in the liquid layer (Press and Ewing [120]). <pi = = [ Jo A(k)Jo{kr) smvrz dk for < z < h (4-172) ^[' Jo f [B(k) sin v.6 to 0. Thus it seems clear that first-mode Rayleigh waves in the period range 1 to 12 sec suffer very great attenuation in typical ocean areas.174 ELASTIC WAVES IN LAYERED MEDIA Rayleigh waves. we have to make use of two different expressions for the potential (p^ for Hquid layers above and below the source. As in the problem of two liquid layers (Sec. The displacements are represented by Eqs. Now. represents the compressional disturbance at z > H ^ h < The of period equation for vanishing A(k) in (4-144) yields the roots k„ Eq. (4-178) which from a compressional H.h by z.^'. we have in this case Jo v\ A(/c. Replacing z by H -\.^l[4k'p. The approximate values of these integrals were evahiated by Honda and Nakamura [68].ill - h)\ dk (4-176) = 2e / Jo(A. (l^. J_ sin e:(K'„) Unhy\^2 - l) sin U.d. whereas the branch hne integrals diminish as r~^. (4-178).r)/v Jo z-~r7T\ ^l A('W ^^^ în v.0i[^k'\v'. (4-147).p.) of (4-13G). C. + (2/.(. A discussion of the period equation has been given earlier in this section. /32 + ip[' v. = /c„2 = w/â and k = A-^2 = <^/02..A LAYERED HALF SPACE 175 Determining the functions A. we obtain the expression given by (4-126) and (4-137) for the disturbance in the first layer from a source in the second. = pSXc'/al pAH Vl .The residues which diminish as r~' yield the normal-mode solutions.p. by po in Eq. and the solutions can again be expressed as the sum of the residues of the integrands and two integrals along branch lines corresponding to the two branch points A. + {2k' - /4)1 cosP:(H - z)\ dk (4-177) = = -2e'"' f Jo{kr)k^-^^ j J.{kr)k^-^^ pô:\2k^' - k%)e-'''''-"' dk (4-178J ^2 -46'"' pô^'ivê-''-''-"' dk (4-179) As a check we may use source z reciprocity considerations upon Eq.-' - /4)"] QOHv.cVa l cos {K„H\/c\/a\ - 3^^ 1) ..(H 1^^ P2 I z) + ^2 v. The integrals (4-176) to (4-179) can be evaluated by the methods applied before.\)M .^ Jfs - l) (4-180) where e. B. I. and p. (Pi the normal-mode solutions can be written in their final form = <Pi — 2 '2t y. S^ as before and using the determinant A(/. c'(2 (4-184) with M = ^|.cVaf .and group-velocity curves calculated for a granitic and basaltic ocean bottom earlier in this section are applicable here.1) cV^l ^2/2 2/2 sec C /ai 2/ ^ — "2 "i C TJ /c„ii^ -2 „^ — VT"1 1 \o!i C -vr COS K„H-^ -o +^ Vi - c7. P2 /32 sin KnH '\/c~ /a\ 1 L a/c'/ C /Os — 1 Vi . the expressions for ipi. The amplitude factors (4-181) and (4-184) determine the relative strength of the various modes as a function of the frequency / = c„/c„/27r. We can now describe the sequence of normal-mode waves as they will . in (4-180). depends on the depth of the source.cVc / 1 + 1 2/2 . The generalization for a pulse in this case was also given by Press and Ewing. and \p2 show immediately the influence of each variable of the problem.c^lal pSl V?/al . The phase. The changes in these potentials produced b y a varying z depth are represent ed by the l ast factor.J 176 ELASTIC WAVES IN LAYERED MEDIA and 2 (^2 /2^ V- 1 i wt — K„r -i)] —kXz •62(0 2 ^2 sin \ Knhx-2 \Q!i 1 - ij exp - ^)\/l 2 0:2 (4-182) _ ^^ •22(0 sin iioot — Kj kJi\-2 — 1 j exp — Kn{Z -H)^ (4-183) where 62(0 = - H2(k„) = .\M pSI Vc 2HpxWl . <P2. sin (k„2\/cV«i — 1). 4-27. ^2 = LoaO are given in Fig. 4-28.c^^DkM . and (4-183)."2 CV^2 + 2^ 1 - 1-M-^ — 1 (4-185) ^2/ the potentials Written in the form (4-180). (4-182). 4-26.1 M p. [204] are reproduced in Figs. Useful curves computed by Tolstoy and 4-29. Additional numerical values representing the conditions for a sedimentary The factor sin (k„/i Vc/«i — 1) bottom (pi/pi = 2.0. 0:2 = \/3/32. a o E o 1 \ o o \ u 1 ^ y' / y y ai / '\1 \ V € \ "^ ^^^^"^ !^| '' > . / / \ / i\ I 177 . C3 L in U.- y^ y^ __^^ '''^ / y y^ y' ^^ . / y *V \ o 1 -^:i r^ ^ i| ) a e3 e / / / / \ / /' / / // . the source (i.e. rapidly at first and then gradually. For t > r/0. At the time t = arrives.1. as/ai = 6. = VS.. 4-30. and the amplitudes also increase. Dispersion curves for a2//32 first five modes. gradually increasing in amplitude to become the predominant waves.998Q:i the high-frequency and low-frequency branches of the group-velocity curve contribute waves which arrive simultaneously and approach each other in frequency until they merge to form a conspicuous train of waves. the normalmode contributions begin. Liquid layer over solid substratum.36) but increase to large amplitudes shortly thereafter. The second mode begins with waves arriving with a cutoff frequency at the time t = r/jSj. the amplitudes of these waves are zero at / = r/ai(j = 4.) At a time t = r/^2 after the initial impulse at immediately after the arrival of shear waves). According to Fig.178 ELASTIC WAVES IN LAYERED MEDIA Fig. which terminates at a time corresponding to propagation at the minimum value of group velocity. and located within the liquid layer can be taken from Fig. 4—27. With increasing time the frequency increases from zero.The amplitudes are zero at the onset and thereafter . to a distant impulsive point source of compressional waves having g(co) = const (a fiat spectrum). {After Tolstoy. The wave amplitudes in the first two modes due arrive at a distant point. pa/pi = 1. 4-30 as a function first of frequency. the Airy phase. In the mode the first arrivals consist of low-frequency Rayleigh waves with very small amplitudes. traveling r/0.998ai a high-frequency train of Stoneley waves with a speed slightly less than that of sound in water. P2 = PI.3. solid substratum./^. = V3.1.6. 0. 0. {After Tolstoy. First-mode dispersion curves for cases • • .A LAYERED HALF SPACE 10 r 179 t3|a Fig 4-28 Liquid layer over «. «x//32 = 0.2. 0.) . 0. waves (7 -^ CO ) arrive.2. 0.1. Liquid layer over solid substratum.3. {After Tolstoy.ELASTIC WAVES IN LAYERED MEDIA Fig. The zero at the onset but increases rapidly there- . Second-mode dispersion curves for cases ai/?2 = \/3. 0. 4-29. traveling ampUtude of these waves is At the time t = r/a^ high-frequency with the speed of sound in water. P2 = pi. • • • .9. ai//32 = 0.) increase as the frequency increases. 4-30. G for an impulsive when p^/pi = 2. It is to be noted that a maximum value of group velocity is also present in the second mode.5 10 Fig.0 2. ^( * (4. .5 ) / / = 2 - / r=2C 2.36.A LAYERED HALF SPACE Fir .and high-frequency branches of the group-velocity curve of this mode approach each other and merge at a minimum value of group velocity. a^/Pi. and the resultant amplitudes show only a minor increase.phase.0 / / 1/ 0. Liquid layer over solid substratum. = Vs. Ordinarily one might expect the large-amplitude waves of an Airy phase to begin here but the "excitation" function Qiik) in Eq.5 / f / r r = 2 oc H y 1/ 1. ending the disturbance with the large-amplitude waves of a second-mode Airj.0.1 y / J \ -^ 0.t 181 r = 2000/7(7 m od«3.0 000 H OHQ y 2 / / T / - Secon d mc de / 2 1.0 y ' ^ A(oo. (4-181) almost vanishes for the value of kH corresponding to this stationary value of group velocity. Amplitude function source. ^2/oii = 1) 3.5. C» 3.5 1 ^ 0.^ - _c/ai t/V«i' ^7 after. For / > r/ai the two arrivals corresponding to the low. - 8 1 d(t7/ai) G(Kn) = eiKn) Sin2 (KnHVcya. P2/P1 = 2. 4-31).15 7= C= Cr First mode 7=0. From the theory just presented a number of important conclusions can Surface Bottom 4.301 7 = 0. Vertical-pressure distribution in liquid for the first and second mode when a2//32 = V3. may 1) factor sin vertical variation of vertical displacecan be easily drawn representing Figures 1). ment by cos {K„zVc'/al (Fig.2 7 = 0. Liquid layer over solid substratum. mode each for of y these variations in terms the first two modes. 4-31.25 7= 0. 02/oti = 3.58 C= Cr Second mode Fig.36 7 = 0. .5.0.182 ELASTIC WAVES IN LAYERED MEDIA (4-180) it be seen that the vertical variations of pressure and horizontal displaceme nt in the hquid layer are determined by the From Eq. The to limited been thus far has The discussion the superposition of the by obtained evidently point is a wave motion at {k„z Vc'/ai - and the - contributions of all modes. 3. 4. These waves increase in frequency and amplitude with increasing time. The velocity of the Airy phase is independent of water depth. bottom.and solid-bottom theory a high-frequency "water" wave traveling with the speed of sound in water arrives. An additional feature of first-mode waves over a solid bottom is the higher-frequency Stoneley wave with c = 0. the amplitudes of waves traveling with the in the bottom will be relatively small. 2. by the depth The frequency of the Airy phase is determined water and the elastic constants of the bottom.and solid-bottom theory the water waves and rider waves merge to form a train of large-amplitude waves which is called the Airy phase. respectively. 4-3). For any given mode and frequency the best location of a of hydrophone is at a pressure antinode. In most water-covered areas where refraction shooting layering in the bottom occurs. the above theory is applicable for all wavelengths considerably less than this thickness (see Sec. 4-31 the vertical location of a receiver for peak response at a given frequency can readily be obtained. For both the liquid. 4-31 we see that the response of a hydrophone sensitive to pressure changes or a geophone sensitive to the vertical velocity of a water particle must vary with depth. bottom made above is the first bottom layer is several times greater than the water depth. For a bottom which can be treated as an ideal liquid was 4-2 that waves having larger amplitudes appear shortly after the arrival of the bottom compressional or ground waves. The frequency of the water wave shows a marked decrease with time. 5. From the vertical standing wave pattern shown in Fig. and the ideal location of a geophone is at an antinode of vertical displacement. The shear waves begin with a limiting or cutoff frequency For a solid speed of comprossional waves which is characteristic of the depth of water and the elastic constants of it the bottom. riding on a low-frequency "rider" wave (see Pekeris [116]). These ground waves begin with a cutoff frequency in a manner analogous to the shown in Sec.A LAYERED HALF SPACE 183 be drawn concerning the propagation of explosion sound over large ranges in water-covered areas: 1. a train of waves arrives at a time corresponding to propagation as Rayleigh waves. With the use of curves such as those of Fig. It is only after the arrival of the first shear waves that large-amplitude waves appear. For a solid bottom. Antinodes and nodes for pressure correspond to nodes and antinodes for vertical displacement (or velocity). shear waves of the solid-bottom theory. For both the liquid. For the liquid and solid bottom the amphtude of the high-frequency waves increases with time. and the assumption of an unstratified indeed an oversimplification. If the thickness of .998ai. is undertaken. Occurrence usually when a hard stratum is found at or near the sea floor They Burg. 4-32 and 4-33: 1. cps 100 Fig. {After Burg. Press. Press.) 10 50 Frequency. Sixteen.155 to 2. and Stulken.to 40-cycle/sec seismograms showing almost pure sine waves corresponding to leaking-mode propagation in water depth of 132 ft. In seismic prospecting in shallow water a surface characteristics. 50 lb of dynamite at 5-ft depth. as illustrated wave has been observed. Large amplitudes and long duration Almost constant-frequency train of waves all in some cases. {After Burg. Ewing. Although a sHght leakage of energy occurs Aîth each reflection from the bottom. 4-32.) 3. 192 ft.405 ft. characterized in of a pattern of cases by numerous repetitions waves Fig. Water depth. apparent mixture of several discrete frequencies in others. having the follomng in Figs. and Stulken [14] gave a theory for these waves. Seismometers at 8-ft depth. stated that waves propagate by multiple reflections at angles of incidence between the normal and the critical angle for total reflection. distance from shot 2. The attenuation is com- . Press. Frequency-response curve on filter setting which on seismogram at right admitted third and fourth leaking modes.184 ELASTIC WAVES IN LAYERED MEDIA Leaking Modes. and StuLken. under the condition of constructive interference. 2. fairly simple pattern of beats in others. Ewing. 4-33. Ewing. there is an automatic gain control on the recording apparatus. "]\Iicroseism storms" occur in the period range 2 to 10 sec. one must modify the theory presented earlier in this section. Eq. Some Aspects of Microseisms. and c/ai = esc 6. For a more detailed description of propagation in a leaking mode. (4-187) becomes «i(2y^ - 1) 4^ It is cose _ ^ _ f . = '^ "'^^l' (4-189) For these leaking modes extremely low values of group velocity are which accompanies them. a simple system of beats would be recorded.A LAYEHEU HALF SPACE 185 pensated so that the recorded amplitude remains approximately constant many seconds after the initial impulse.l. Although . • • • (4-187) If we use the relations k = 2x d/lt. These storms last from a few hours to a few days. If two modes pass the filter. one plicated patterns mentioned above. nodes and antinodes occur in the water. at various depths and the contribution of each mode to the seismogram will depend greatly on the depth of the shot and of the detectors. during which time the amplitude of motion gradually rises far above normal and then gradually decays. may As may be expected. The phase velocity and the group velocity can be approximated by assuming infinite density for the bottom.-^ r4-iss) ^^ ^^^^ ai sin 6 easy to prove that the group velocity U = = r/t approaches 0) zero as the angle of incidence approaches the normal (d = and the frequency approaches the value /. 4-33. It is also seen that many modes may be propagated simultaneously. as illustrated in Fig. and several modes together would produce the more comgiven water depth. Equation (4-154) then takes the for following form: tan or ( kllJ^ - 1 ] -^ oo (4-186) kHJ-. 2. because of the automatic gain control mentioned above. (4-189) if the filter allows only a single mode to pass. Oi\ 1 = ^— 2 sin TT n = 1. limited principally significant despite the increased attenuation by the type of wave filter used in the recording apparatus. The useful sensitivity of most seismographs is limited by background oscillations called microseisms. Thus for a observe a single wave train whose frequency approaches that given by Eq. of wind and pressure fluctuations within the storm. in opposite directions. of water depth. although it was a second-order effect. They concluded that. Further waves were stimulated by the conclusion of Bernard [5] that the period of microseism oscillations on the African coast is half that of the generating sea waves. it was adequate to produce the observed microseismic disturbances. NATURE OF SOURCE. In agreement with the finding of Miche they found a pressure fluctuation at the ocean bottom having half the period of the ocean waves and made the first attempt to calculate in detail the amount of energy transmitted to the ocean floor and thence to the point of observation ashore. Banerji [2] proposed that gravity waves on the ocean transferred energy to the sea floor. The problem of microseisms may i.. MODE OF TRANSMISSION OVER OCEANIC PATHS. Gherzi 2. Longuet-Higgins and Ursell [89] and studies of the transfer of energy from ocean surface Longuet-Higgins [90] presented a detailed theory for the effect at the ocean bottom of interference between two similar wave trains traveling [94]. 1. conveniently be divided into four the role of the ocean in the transfer (2) parts: (1) the nature of the source. of energy from the atmosphere to the earth. [49] believed barometric pulsations within the storm were able to transmit the necessary energy. is available for them which can explain all all The sole points on which agree are that they are generated by the action of storms at sea and that they affect areas of continental dimensions. is It is natural to suggest that the energy of microseisms transmitted from the source to the . of storm position. and of the effect of such parameters on the period and amplitude of the resulting microseisms. See Refs. an idea which Scholte [152] tried to revive by including the effect of compressibility of the water. 100 and 113. After more than 50 years of observations. the mechanism of transmission over oceanic paths. (3) effects at the continental margin. An early theory (Wiechert [208]) which still receives some support held that swell breaking on steep coasts introduced microseismic energy into the solid crust. no theory the observations.186 ELASTIC WAVES IN LAYERED MEDIA studies of storm microseisms have resulted in hundreds of papers during the past 50 years. there is still lack of general agreement about such basic observational data as the role of ocean waves. Deacon [22] noticed the same relationship between microseisms at Kew and sea waves on the north coast of Cornwall. and (4) the type of propagation over continental paths.e. It is not necessary even to summarize the history of this subject here. as two complete volumes have been devoted to it. Perhaps the major cause of the diversity of opinion is that most observations have been of made in latitudes v\^here the general movement weather is in one direction and the sequence of events during a storm passage from land to sea differs radically from the sequence for a storm passing from sea to land. Gilmore have stated that storms He has pointed out that microseisms originat- Canada propagate to great distances over the United States and Canada and decrease noticeably only after passing the Rocky Mountain area. A profound geological change occurs at the continental margin where the crustal thickness increases from about 5 to about 35 km. 1308. and 55] that microseisms geologically disturbed areas. Press 187 and Ewing [117] modes of wave propagation diseussed and Longuet-Ifiggins [90] implicitly assumed that the propagation could be adequately represented by the theory for a homogeneous liquid layer over a homogeneous soUd half space. Dinger and Fisher [24]. 4-5 and Fig. It was first noted by Gutenshow marked attenuation in berg [58. He states that in all instances ing off the Atlantic coast of where the microseisms are propagated over long distances without much loss of energy. This is in marked contrast with the abundant energy in the same period range transmitted over continental paths. Several observers report opposite conclusions. Perhaps this difference of opinion is due to uncertainty about which portion of the storm produces the microseisms. Unfortunately. For example. stations and source are on the same geological unit. and he has called attention to several similar barriers in Europe and in the West Indies. The results of Donn [30] and Dinger and Fisher [24] support this idea.3 produced clear phases called Lg and Rg (see Sec. and Hubert [51] and Whipple and Lee [207] all well beyond the continental margins produce significant microseisms. an earthquake in California of magnitude 5. no such waves were found where any part of the propagation path crossed an ocean area beyond the continental margin. It strongly supports the conclusions of Ewing and Donn [42]. Although the cause of the spectral gap has not yet been found. It is therefore a logical extension of Gutenberg's views about barriers to expect that the transition between ocean and continent will introduce extreme attenuation in microseism waves. the gap in Rayleigh-wave spectra for oceanic paths discussed earlier in this section apparently eliminates the possibility of applying this theory to microseisms. They showed that the hurricane microseisms at stations on a continent increase greatly in amplitude as soon as the storm touches the . 4-56) with periods 2 to 8 sec at Palisades. and Carder [18] that ocean areas beyond the continental margins transmit microseisms very poorly and that large microseisms occur only when a portion of the storm (or its swell) reaches shallow water. the radius of the storm area being often comparable with the distance of the center offshore. 3. EFFECTS AT THE CONTINENTAL MARGIN. despite its utility for longer-period earthquake surface waves. the gap is clearly of the greatest importance for microseism studies.A LAYERED HALF SPACE continental margins by the normal in this section. p. Despite careful search for earthquake surface waves in this period range on seismograms from many coastal stations and from Honolulu and Bermuda (Ewing and Press [43]). . and Donn [31] has shown that microseisms which are initiated by storms reaching the Pacific coast of Canada may be identified at Palisades by the appropriate Rayleigh-wave particle motion. It is seen at once 1000 fm Vertical exaggeration 4:1 I . MECHANISM OF PROPAGATION OVER CONTINENTAL PATHS. This means that either (1) there is poor generation there is and transmission of microseisms in deep water or (2) extreme attenuation of microseisms at continental borders. . 4-34. .188 ELASTIC WA^1:s in layered media continental margins. 4-34. I . 100 100 Distance in 200 300 400 500 kilometers from Seven Mile Beach Fig.) and gravity data. The counterpart of microseismic propagaberg [60]). . This is the one aspect of the whole microseism problem which seems to offer no difficulties at the present time. Structure of continental margin deduced from seismic {After Worzel and Shurbet [212]. Disturbances from the atmosphere or from the ocean entering this crustal layer in the transition zone will find that the sloping boundary constitutes a Lummer-Gehrcke plate strongly favoring propagation toward the continent rather than toward the ocean. The fact (see Sec. continental borders strongly supports statement The observations of amplitudes smaller than expected which have led to the idea of barriers to microseismic waves are due to the two mechanisms listed above but the relative importance of the two is not clear. importance for microseism studies. I . . Transformation of surface waves incident upon the continental boundary from one type to another 4. I .to 12-sec surface wave trains from submarine earthquakes do not cross 2. that transmission across the boundary in either direction is impossible for any wave type in which the crustal layer acts as a wave guide for normal modes. Murphy [95] and Gilmore [50] have pubUshed additional evidence about the barriers. 4-5) that the characteristically continental phases Lg and Rg and 6. The details of structure of a continental margin as revealed by seismic refraction and gravity studies are shown schematically in Fig. . or both. is of greatest The efficiency of transmission of microseismic waves across continental areas has been considered remarkable since the early days of seismology (see Guten- Carder [18] has made additional studies supporting this point for North America. . . Ref. The period equation similar to its those discussed in previous sections can be readily obtained from general form derived in Sec. Wave propagation in a semicovered by a solid layer of uniform thickness was [12] for first infinite solid studied by Bromwich steady-state waves of length large compared with of length 3.seisms. From this it follows that in many cases a solution for harmonic plane waves is adequate. In the preceding sections we considered the solution of problems of wave propagation from an impulsive point source. He was led into this study in an attempt to explain the duration and complexity of earthquake waves. resonance of crustal columns. once microseism energy enters the continent. even drastic modifications of the fundamentals of the classical theory of wave propagation. The importance of Lg and Rg for microseism study [125] is that they demonstrate the existence of a very efficient wave guide in the continents. 4-5. in search of a theory of earthquake the layer thickness. wich to include waves surface waves.A LAYERED HALF SPACE 189 tion across continents is probably represented by the Lg and lUj earthquake phases described by Press and Ewing [125] (see Sec. whereas the Rayleigh-wave motion of Rg matches that observed for longperiod microseisms. that the principal conclusions waves may be obtained directly from the characteristic relation between period and phase velocity which appears in the same form for each physical system. Solid Layer over Solid Half Space. It all is now clear that layering is responsible for practically the observed effects. Since that time numerous appeals have been made to imperfections of elasticity. Love (Chap. Rayleigh Waves: General Discussion. Thus. = Ae'""-'-'-'"' + Be''"'-"''^"' (4-190) ^2 = Fe' (u ( —kx) — Vt'z propagating in both media. and we shall restrict our discussions to this simple type of problem in the for surface following pages. . scattering. it can spread over great distances with very little loss. as Love suspected. It can also be derived by assuming that there are plane waves <p. regardless of the source of the wave. however. 4-5). A similar method is directly applicable to the problems treated in the remainder of the book. Both phases meet the requirements of period and efficiency of propagation. The applications have demonstrated. 4-8 for n — 1 solid layers overl3dng a solid half space. The orbital motion of Lg is not unlike that found in short-period micro. 26) extended the work of Bromcomparable with or small compared with the layer thickness. 5e^'^ iDe'^'". .190 ELASTIC WAVES IN LAYERED MEDIA Substituting these expressions in the boundary conditions (4-4) and (4-2) written for plane waves.2kv.viiCe-"'" + v{iDe'''" = -kEe-""" . and dx d(pi .)iD = -kAe-''" . dz d\pi dx d(p2 I dz (4-192) d\l/2 .Ae-"" 2kv.kl)B + 2kp{iC .B + {2k^ .' - yfcÔAe"^''' + {2^ = kl)Be''" + 2A:f'(zCe-''^'' ^ Ml [{2k' - kl2)Ee-'"'' + 2kp'2iFe-"'"] The six variables Ae""'".2kv[iD = 2kv. iCe"'".kBe"" .kl.x)2 we obtain six equations: i2k' .kl.A .Ae-''" + v.)iC + {2k'' .Ee-"'" - kiFe'"'" (4-193) + {2k' = Ml kDiCe-"'" + {2k' (2fc' kl)iDe"''' kl2)iFe-"'^ 2kv[iDe'''^ 2^ kv2Ee-''" + ^ Ml - (2A.Be'"" kiCe-''" - kiDe"'" = -v. Tj dz dx (P. Ee'"'".kl)A + i2k' .) 2 dx = (P. different must have values equation from zero.x)l = (P.. and iFe~'"''' and therefore we obtain the period A = where the determinant can be written in the (4-194) form shown in (4-195).z)l dz (v.vîFe-""" -p.Be''" 2kv. fii I (N I ^ (N V ^ => Oi ^ .A LAYERED HALF SPACE 191 TjH :l.1 a. ^1 i a. CI (N 1 1 ri? -y (M (N -ii I I I <N . ri^ a.^ 1 N <Q.l a.1 a. ^Si 1 1 . e-'-"" + D. + D. - kl)A^3 D.192 If ELASTIC WAVES IN LAYERED MEDIA we put Fik) f{k) = (2k' - kl. . -k 2k' . -De -4kp[(2k' - kl)A.Y - 4:k\y. D. kl. (4-199) -2kv.kl. The factors A. If the columns are combined in another way.3 v[ • ' = 2k' Pi -k 2k ' = -2kv.. • • - kl ' ' -2ki'[ • ' -k A24 -k • = 2kv. and = -/A23 FA34 A12 is the subdeterminant of the fourth order shown in (4-195). . Pi -k A34 -Pi . . the last two columns being identical for all: -k Ai4 -v[ ' • -K A. + Dê-'"''' + (D. ' ' = 2k' — Vi 2kv.{2k- D.kl k]. - • A2S = 2k' -k — Vi 2kv. = = = /Ai4 D. (4-196) 'ik'py^ = (2e . the determinant (4-195) yields the period equation in the form discussed by Sezawa and Kanai [169].. -k 2k' . are other subdeterminants of the fourth order formed by the elements of the last four lines of A. Equation (4-194) will be discussed later in this section. (4-198) D. 4kp. — 2kv[ kffi • -k 2k' . (4-195) can be represented as the sum Apx+v^')H [D. ' • 2k' • • 2k' - kl.kl k ^1 2 • • • - -2ku I - 2kv[ • • Using the expression (4-197) we have the period equation (4-194) in the form obtained by Newlands [105]. 2kv[ -k 2k' • - kl.e -2(vi + Pi')Hi (4-197) = = FA.)e- + where D. where exponentials are replaced by hyperbohc and .kiy + Eq. X = M2 Ml ^2 k"^ Y = ^. cos r . Ref. +2l 172 A" sin rii/ — Y — r^ cos r. = = V2 (4-201) = (^^J'i)^ = - v'2' the period equation in the form written by Lee îi72 — â^?! = (4-202) where ^1 = 2 ')[ X cos r^H -\ F sin mriH r. 193 is A form often useful for computation that given by Lee [85].»] = o 12 I - ^ "'I*' 7^ k W cos Sii7 + — Z sin Sii/ s. cos s^H (4-203) = 2 ^1 k'" T k W COS TiH H r. are obtained when k' < kli < kli and k^ > k% > provides an implicit relation between phase velocity Equation (4-202) klic and wave number k. Two branches of this function . Z sin r^H + 2| X sin Sii7 i?! Y s. r. 20).. W sin SiH + 2| T k ^2 Z s. Z.A: (4-204) sin SiH + 'i Positive real values of W sin riH — —Z r.A LAYERED HALF SPACE trigonometric functions. It If X. 2l^' Ml 3. through the dimensionless parameters c//3i and kH.// =(2-S A cos SiH -\ —F S2 y. Y. W are expressions introduced by Love (Chap. + I^Pl (4-200) M2 A*2 Hi kff2 '"pa n k K fe-0 = kai ^^1 W Ti si = 2 and ^f Si = (ivi) — — k k^ = = k k^ is - ka2 k'i. c may be obtained as a function of the period T through the relation T = {2irH/î)/{kH -c/î). cos s .»] s. Alternatively. is given here for reference. This is the branch which all is relevant to the propagation of earthquake Rayleigh waves. does the phase velocity approach the speed of Rayleigh waves in the substratum. while the zero of the second factor represents Stoneley waves at the wave types These results provide a check. used in previous sections for deducing the group-velocity curves and some characteristics of seismograms from dispersion curves such as those presented in Fig. by Sezawa and Kanai Kanai (see Sec. For c < I3i and kH —^ 00 the asymptotic form of Eq. Sezawa and Kanai [171]. M22) of the Ml and M2 branches were made by Tolstoy and Usdin [203] for the case pa/pi = 1-39. The zero of the first factor represents Rayleigh waves at the upper surface of the layer. These curves are shown in Fig. Under the very stringent conditions necessary for the existence of Stoneley waves at the interface between two solids.147. and q. whereas the M2 branch leads to the opposite type. Further discussion of the roots of the period equation may be found in interface. corresponding to a cutoff frequency below which unattenuated propagation does not occur. . In other modes the velocity approached at the long-wave limit is that of shear waves in the substratum. by substituting in the expressions for vi and vi used in computing roots of the period equation (4-202). At the long-wave limit only in Mn.194 ELASTIC WAVES IN LAYERED MEDIA occur corresponding to the M^ and M2 (see also type of propagation first described Sato [141] and later Tolstoy and Usdin [203] have suggested that these two wave types correspond to the symmetrical and antisymmetrical modes for a free plate [171] [81]). first • two modes (M n. that the particle motion at the surface for the ilf. As in the previous cases. It may displacement the values of be shown. giving the expected wavelengths small compared with the layer thickness. Although the period equation (4-202) is more complicated than that for a liquid layer on a solid (Sec. At the short-wave limit (kH -^ co) the phase velocity for the first mode of the Mi branch approaches that of Rayleigh waves in the layer. phase velocity c» approaches that of shear waves in the layer as kH > The methods . as pointed out by Love. — .i//3i = az/iSg = \/S. 6-1). The cutoff frequency increases with the mode number. roots of the period equation is limited. /â//?! = 3. and KeihsBorok [82]. 4-35. and for each mode there is a least value of kH. (4-202) becomes factorable. the first mode Calculations of dispersion curves for the • • . branch is the retrograde elliptical type normal for Rayleigh waves. Keilis-Borok found that for a given w and a given layer thickness the number of real Scholte [156]. 4-35 may readily be applied. 4-4). it may be treated by the same method to determine the number and location of the roots. for Neumark [101-104]. there would be an additional mode in the M2 branch for which the phase velocity approaches the speed of Stoneley waves as kH -^ 00 For all other modes. of the Ml branch. <Ja. I \ \ \ \ \ 1 II .---'' II II ' / / Jy^ ( y^ |ci \ ^1 ^^ 1 \6 ^\^J ^1^^^^ fc 195 . 1 \ 1 1 1 1 1 a n \ \ 1 ' 1 ' 1 ( 1 1 ' 1 1' 1 8 '' '1 1 ' ' 1 / 1 ::: 3 1 [ 1 " // ' / ^ / 1 ^ / y \ \'i-\ '\ s. In accordance with the theory of the propagation of Rayleigh waves in layered media. From these studies we have selected the last two to present in Fig. Propagation of Rayleigh Waves across Continents. This large path is required because the dispersion is much weaker across continents than across oceans. among others. independence from error due and origin-time data. Sezawa [170]. and Oliver [128]. Brilliant and Ewing avoided the requirements for long continental path and accurate epicenter location by utilizing waves from a shock in the Southwest Pacific which crossed the west coast of the United States at normal incidence and were recorded at six stations distributed across the United States on a suitable great-circle path. one requires an earthquake of magnitude about 7. and Press. These results are consistent with each other. is equivalent to having a pure continental path of great length. Because of a variety of difficulties. that they have adequate response in the period range 5 to 75 sec. It is noted that several maximum and minimum values of group velocity occur in the higher modes. The advantages of this technique are (1) to normal inaccuracies in epicenter . resulting from lengthening of the train over the oceanic segment of the path which. and the conditions of the experiments were the most favorable of any for yielding data about purely continental paths. It is further desirable that the seismographs have three matched components so that the Rayleigh waves may be positively identified by particle motion. in effect. 4-36. 4-36. In no study have these conditions been met adequately. It is straightforward to determine the group velocity for each selected period across this spread of stations and to demonstrate its constancy across the continent. For a number of selected periods the arrival times at each station were read. attempts to obtain a precise dispersion curve from empirical observations have not been very successful. Brilliant and Ewing [11]. which produces Rayleigh waves over a broad range of periods and has a well-determined epicenter situated at one end of a chain of seismograph stations well distributed on a great-circle segment about 70° in length. The results are plotted in Fig. (2) improved accuracy in determining periods. Figure 4-37 shows arrival time as a function of period for all stations. Carder [17]. and that the path be free from major crustal irregularities. Ideally. Ewing. it is observed that the velocity of Rayleigh waves across a continental area is dependent upon period.196 ELASTIC WAVES IN LAYERED MEDIA the phase-velocity and group-velocity curves meet at both the low-frequency and the high-frequency limits of kH. Observations of Rayleigh-wave dispersion for continental paths have been made by Rohrbach [133]. and (3) independence from error inherent in other techniques involving mixed paths and from uncertain correction for oceanic and transitional segments. Wilson and Baykal [210]. Gutenberg and Richter [56]. minutes Fig.5 * New 10 September 1954 05-44-05 York (Brilliant and Ewing) D Mantle 2. 4-36. /?! 3.0 Rayleigh Wave (Ewing and Press) 10 20 30 Period in 40 seconds 50 60 70 Fig.25 /Oi 3. Method of determining dispersion of continental Rajdeigh waves. 4-37. /sec ^2= 4. o A Algeria to Natal 9 Algeria to Natal California to New York 1 March 1955 September 1954 01-04-37 2.5 P2= 1.51 km. Period is plotted against arrival time for six stations on great-circle path across the United States.5 197 4. Observed and theoretical dispersion of continental Rayleigh waves. 15-42 -46 -50 01-08 -14 -54 -20 -58 16-02 -25 -06 -32 -10 -14 -38 -18^ -44 Arrival time.) .0 35 km. hours. (After Brilliant and Ewing. /sec 3.68 km.0 *$x^ ++^ X Yukon to Palisades.A LAYERED HALE SPACE 4. A theoretical curve can be derived using Eqs.198 ELASTIC WAVES IN LAYERED MEDIA Press. The inverse dispersion present in this phase is clearly demonstrated for the first time on this seismogram. 9.51 km/sec. indicates an identical crustal structure for the two continents. 1954.25 times that of the crustal layer. 4-38. with shear velocity 3. Periods for a series of arrival times are read The dispersion data thus obtained are [11]). The point at 44™45^ after shifting the origin time. and the beginnings of the Rayleigh-wave train. 4-36: limit of 1. The dispersion from these seismograms could be measured with great precision because the path is longer (7. The curve assumes a homogeneous crust 35 km thick. The following conclusions may be drawn from Fig. 10. Poisson's constant a is taken as 1/4. is taken minimum value of group The surface waves arriving during the next 10 or 15 min after the Airy phase are interpreted as due to scattering and reflection from inhomogeneities in the earth's crust. wave number. along with a group of waves which is interpreted as an Rg phase reflected from the continental margin. overlying a homogeneous mantle with shear velocity 4. Time is marked in minutes after the origin time. and the Rg train (to be discussed later in this section) are indicated. to represent an Airy phase corresponding to the velocity of continental Rayleigh waves. The method Press [41]) of deducing dispersion from a seismogram (Ewing and involves reading times of zero trace deflection and plotting these against as slopes of the resulting curve. plotted in Fig. and Oliver studied the Rayleigh waves from the Algerian tremor of Sept.1 km/sec.68 km/sec and density 1. Union of South Africa. difficult to With the arrival of the Lg phase. The Rayleigh-wave portion of both seismograms is reproduced in Fig. it becomes read the ordinary Rayleigh waves. . in which the wave period decreases with time. but they can be analyzed by with great amplitude. These were recorded on a long-period Columbia University vertical seismograph installed at Pietermaritzburg.890 km) than any which has been available for a longperiod vertical instrument and is free from such obvious crustal anomalies as mountain ranges. Rg begins abruptly may be seen clearly on the seismogram of the aftershock. Emng. as from the main shock to the aftershock. the Lg train. and the aftershock of Sept. just prior to the arrival of the reflection train. The Rayleigh-wave train for the main shock clearly shows the normal dispersion. reported in the past. 1954. (4-202) and (4-94). 4-36 along with the data for the United States path (Brilliant and Ewing Also plotted is a point indicating the short-period mantle Rayleigh waves (Ewing and Press [46]). Jeffreys' calculations [79] were modified so that the elastic constants agree more closely with the most recent determination of crustal structure from explosion and rock-burst data. The remarkable agreement of dispersion data from Africa and North America. in which the discrepancies are less than 0. 2 2 5:5 .^ \>u ^ -S 's-t ^ o ^ « ^ 5 3 ^ .= C bC o "^T: =3 CJ Pi'o 2 S fe -+3 199 . <D 02 O m M a VI a> 3 J 2i -u> o — S o u 7* 03 c3 ^ ->l „ ^ ^ 3 J3 o3 r^ ^? > '^ >^ o -* <0 .O a o o. In many regions the near-surface layering consists of a zone of poorly consolidated sediments overlying more competent beds with higher velocity. The method of Dobrin et al. the group velocity decreases with increasing period in the range 70 to 225 sec.2 km/sec. it has been experimentally verified in the study of mantle Rayleigh waves (Sec. as do the experimental points.7 km/sec 3. into this problem. 4-39. [28. In some cases three-component detectors were placed at various depths in a borehole. and we cite some of the results of Dobrin et al.200 2. Roll. is needed. one which includes the is Ground a velocity gradient in the crust as well as the mantle. A small-scale counterpart of the preceding problem that of the "ground roll" frequently encountered in seismic prospecting for oil. because of the velocity gradient in the mantle. Theoretical curves including the effect of heterogeneity in the mantle are not readily available. above the theoretical The observed minimum group velocity falls at a shorter period than that indicated the theoretical curve. since there velocity. we can interpret this discrepancy as an effect introduced by an increase of velocity with A new effect of theoretical dispersion curve depth in the crust. ELASTIC WAV'ES IN LAYERED MEDIA Although the is over-all fit of experimental data with the theoretical reasonably good. This effect may be explained by the known increase of velocity with depth in curve the mantle. A number of papers appeared on this subject. A sample seismogram is shown in Fig. It is important to note that the discontinuity in compressional-wave velocity at the ground-water table does not enter low-velocity. At least at Rayleigh waves It is seen that the individual crests can be followed across the entire . one distance a three-component detector is placed so that the may be definitely identified by the orbital motion of the earth particles. there is a tendency for the observed points to fall below the theoretical curve for periods greater than 38 sec. 29]. consists of detonating explosive charges at the surface and at varying depths beneath it and recording the resulting seismic motion with a spread of detectors spaced at intervals of 50 ft and extending to a distance of several thousand feet from the source. Since the velocity near the top of the crust is fairly well estabhshed in body-wave studies. We therefore conclude that a maximum value of group velocity occurs between 40 and 70 sec and falls below the theoretical curve of Fig. now fixed by explosion seismology studies. However. in the crustal layer is by would occur if the average velocity higher than that assumed in computing the theoretical Both these effects curve. lie In the period range 18 to 30 sec the experimental points curve by amounts ranging up to 0. Previously the adjustment of the theoretical curve was made by decreasing the velocity in the outermost part of the mantle well below the value 4. 4-36. 7-4) that. is is no associated discontinuity in shear and the shear velocity the principal factor determining the velocity of Rayleigh waves. S 'ri "^ ^-^ 1^ 2 "S I'd ^ S^ ^ -5 ^ "*" c o s 0) 201 . — -1 -J> o '—^ -fci t^- lO '^ 5 o lO c: " 1^ H 8 g = s . ._ 0^ ^ > S O N ( .2 2^ .H i 5 -A-^ "2 J^ £ 3 02 j^ ^ I Qj o tC ^ 'v^ > tn > t. 9 '" .-i ^ /^ *-> -U) -^ -^ '. l^ ^ >.> o •n *" . that the source generates a broad spectrum. as given the empirical curves earher in this section. A fairly good agreement with the theoretical curve.19 sec.17 to 0. as would be expected from an explosive source. the wave motion is not always 1800 1600 1400 1200 1000 800 600 400 200 300 Wave Fig. They obtained is represented in Fig. ELASTIC WAV-ES IN LAYERED MEDIA permitting direct determination of phase and group velocities. applied the theory of Rayleigh-wave dispersion. 4-39. {After Dobrin. It is further seen that the dispersion is strong. 4-40. Dobrin et al. for periods 0. 4-41. 4-40. Although ground roll consists predominantly of Rayleigh waves. Rayleigh waves obtained from seismograms Simon. shown in Fig. length 400 in 600 feet Group and phase velocities of shown in Fig.) . and Lawrence. and that the data are far more complete than is ever likely to be the case in an earthquake study.202 detector spread. using values for velocity and layer thickness found in borehole surveys in the area. Simon. Observed dispersion curve (at the station SWX-3) compared with theoretical curve based on calculations of Sezawa for a layered solid with the indicated characteristics. probably determined by the cutoff frequency of the recording equipment. = 800 ft/sec ft/sec 600 // If (32= ^^2 1800 /î 500 400 / 0. large-amplitude arrival.200 Period in 0. it is probable that they are primarily controlled by the very low rigidity of the bottom sediments. in Fig. Their frequency is well below any used 1200 1 / 1100 iÔb served 1000 at SVVX3 / / 900 j / / / 800 / / / 700 / /^The oretical / cl rve for case: / / / / / / 60 ft /5. A low-frequency. (After Dohrin. and Lawrence.) .175 0. Although a detailed investigation of these waves has yet to be made.225 seconds 0. the lowest frequencies present being 1 to 2 cycles/sec. has occasionally been observed on seismograms from explosions in shallow water (Worzel and Ewing [211]j. Often dispersion is evident.A LAYIiRED HALF SPACE as coherent and regular as that significant transverse 203 shown It is components. with group velocity as low as a few hundred feet per second.250 Fig. 4-39 and often contains suggested that heterogeneity and aeolotropy of the upper layers can account for these phenomena. 4-41. 0 d (/I 2^ Mocie 2 / / Ii j/^V. A number of dispersion curves calculated by several investigators for various values of the elastic 1. (A/ter Haskell.4 1.204 ELASTIC WAVES IN LAYERED MEDIA 4. Table 4-1. Curve 1.^ in seismic-reflection surveys. . Fig.4 1 2 3 5 7 10 30 50 70 100 200 500 1000 T.60 km. case 5.38 km.6 1.39 km/sec Ex = 13.— • " ^ / / - <> / P2I/Pl=l M2/M = 3 0.2 y ^ ^ *w^ / 20 1 ^^^ V 2. J. and their large if amplitude can cause serious distortion of other seismic signals amplifier stages ahead of the filters are allowed to become overloaded. Theoretical Rayleigh-wave Dispersion Curves.2 c 1. Ux = (see Table 4-1).60 km. Rayleigh-wave dispersion.8 2. 4-43. curve 2.) .6 4 5 6 10 Fig. {Ajler Kanai..4 4.8 4. / 3 y ^ ^- -- ^ 3. 4-42. Group velocity of Ravleigh waves for /3i = 3.6 ^ 3.8 •v / 0. Curve 3.î = 28.0 ««. ' y '^-' 3 / U ^-.. 13. sec. 4-47 2 1.449 1. Since the first Love Waves: General Discussion. 4-42. long-period seismographs measured horizontal motion only.000 0. Table 4-1 identifies each dispersion curve.871 curve 1 Haskell [62j 2 1. Tolstoy and Usdin [203] 1 2 1.060 2.000 Fig.A LAYERED HALF SPACE 205 constants of the two media are given here. 1. curve 3 Jeffreys [79] 1 Fig.732 2.309 1. the presence of large transverse Table 4-1. 4-48 2 1. Fig.828 1 Kanai [81] 1 2 1.084 Fig.810 2.236 1 Kanai [81] 1 Fig.732 2.916 4-35 3.110 1.000 .000 1. 4-44 2.370 1 .440 1. .190 1.732 3.472 1 Kanai [81] 1 Fig.000 1.732 1 Kanai [81] 1 2 1.333 1 250 Fig.732 3. 00 4-49 2 1.370 1.732 2.000 .732 7.000 1.000 938 1. and cites the references and the figure number. 4-42. 4-42.620 2. Computation of Rayleigh-wave Dispersion Solid Substratum in Solid Layers over Case Reference Haskell [62] Layer i OCil&X ft/ft 3l7pi II J E.732 1.000 .873 1. 4-36 2 1.414 1 1 Wilson and Baykal [210] 1 Fig.440 1. gives the elastic constants.370 .810 2.390 00 Sezawa [158] 1 1 CO Fig.000 1.732 5.898 Fig.203 1 140 Fig.000 . Kanai [81] 1 4-43 2 1.732 4. curve 2 Haskell [62] 1 2 3 .110 .451 1.000 1. 4-45 2 2.167 . 4-46 4.110 1.560 Fig.106 .440 1 1.810 1.147 1 .746 1. 2 1. case Table 4-1.5 1.il^2^1 =-20 0. Rayleigh-wave dispersion.6 V / Pi/Px = 1 M2/M1 = 5 0.-.6 2.' 2.4 c y X / "^^^^ ^ .6 V / y / ^ 1.8 0.0 1. 4-44. .0 =^ \ \ \ / u Pi / / 0.4 2.— y^ ' - / / / / / / y 1. case 6.8 0.y 10 Fig.. 4-45.4 / / / 1 1. Rayleigh-wave dispersion.6 1 1 Pih -i.8 1. Table 4-1.2 2.0 / A c ^ ^ y y y 1.0 / ^V \ / V 0.4 1 2 3 4 5 6 10 2iT/kH Fig. {Ajier Kanai.2 012345678 27r/A:H 7.) 206 .4 V 1 \ / 0.) 2.2 1 1.8 1. {After Kanai. 6 1 P21'Pi=l. Rayleigh-wave dispersion.8 U /3.0 HALF SPACE 207 1. 2 3 4 5 27r//jH 6 7 8 10 Fig. Table 4-1. Table 4-1.A LAYERED 2. Assume that dis- 1.0 > V / / \ / / / 1 / ^ ^1 0. To derive the period equation 3.8 \ \ 0. 1 M2/M1 = 8 \ 0. Ref. 4-46. in the "main tremor" was one of the first estabUshed facts of It was not until 1911 that an explanation of these waves was provided by Love who showed that they consisted of horizontally polarized shear waves trapped in a superficial layer and propagated by multiple total reflections. (After Kanai. we follow Love's original discussion (Chap.) .8 1.4 1 / ^ . case 9.0 -^^ 0. (After Kanai.*-- '"^ ^^3 2 2ir/kH Fig. Take .6 / r c y ^ ^^ / / / ^ y / 1. with the axis in the direction of all propagation and the z axis vertically downward.M2/Aii=1. case 8.) components seismology. 4-47.2 1 - P2/P i=l-14.65 c 1.4 1. Rayleigh-wave dispersion. . 26) and consider simple harmonic plane waves.r the origin of coordinates in the interface.2 1. 1680 .90 < y // / \ / ^ 4 0. 1400 .85 2 6 8 10 12 14 16 Fig. Rayleigh-wave dispersion. /32=6000ft. Fig. 1960 . Table 4-1. {Ajter Wilson and Baykal. 4-48.) .05 1./sec. I 18 9 4. case 12.5 2. (A/ier Dobrin et al.7 1. case Table 4-1.6 1. Wavelength in ft.4 /in ops 11. 4-49. [29].00 i 1 0.208 4UU0 ELASTIC WAVES IN LAYERED MEDIA 3000 Surface y^ ^ y^ Phase velocity ^^ H=280ft. Rayleigh-wave dispersion according to the theory of Sezawa./sec.95 0./ 840 I 1 t 1 1 8 cT/H 280 I I I I 560 I 1120 . / 2000 y 1000 Group velocity ^ 1 2> I .2 3.) 1.2 2.9 1.1 1.2 2. /3i=1350ft. /?].)i - Be"^'" = = {'Pzy)2 at 2 = leads to )Uifi(A î B) = -M272C ?>2 (4-209) gives The continuity of displacement = at 2 = A +B = C (4-210) . surface 2 and be continuous. B. Making use of solutions of the wave equation in the form (2-7).A LAYERED HALF SPACE 209 placements are independent of the coordinate y and that the time variations are given by the factor exp surface (Fig. we write for the displacements (4-205) (4-206) where ^^ = VI - ^ -^^ ^ VI - ^ (4-207) In order that the energy of these plane waves be confined to the superficial we require that c be less than ^2. The three constants A. that is. The plane z = The equations of motion (1-I3j reduce to (V^ -|and to (V^ = for the layer + kl2)Vi = for the substratum. ^2- 1^2 Fig. ^1 M2. which require that the stress p. kli)vi —H represents the free (tco/). (1-11) the condition p^„ = at 2. Thus by Eqs. 4-50). Notations for a layer underlain by a solid half space. that 72 be a positive imaginary number. and C are determined by the boundary conditions. 4-50. together with the displacement at the interface = 0. // Ml. = —H leads to (4-208) Ae-"^'" and (Pjs.y vanish at the free layer. z ^ M2f2 = ^ Vl . modes correspond to 0. and it may is be readily shown that when ^2 < &\ no relevant solutions exist. The existence of an infinite number of modes follows from the periodicity of the tangent function. • . (4-206). 148]. kjiH -^ 0. is In contrast with the problem for two liquid layers.Si < c < ^2. 1. 7r/2 for the liquids.. and larger periods are is excluded (for unattenuated propagation). 4-53. yet the product —» 7r/2. may be treated according to methods used in Sec. Generahzation of Love-wave theory for an impulsive line or point . and (4-212) that the 2. the Airy phase occurring at a period of about 20 sec when H = 35 km. kH approaches zero. 71 —> 0. 27r. (4-78). It and from the last seen by (4-212) when c -^ ^2. 4-52.210 ELASTIC WA^^S IN LAYERED MEDIA if This system of linear equations has solutions different from zero e-"''" -e"''" A = Mifi 1 — Mifi 1 M272 = (4-211) -1 or . 4-51. in addition to a calculation for the second mode. • • • nodal planes within the layer. = . We note that real roots occur when • . Stoneley [197].H . the group velocity has a minimum value. Stoneley's results. It is easy to different show from Eqs. and Kanai [81]. Eq. tan ki. In the case of liquids. In the Love-wave case there no upper limit to the period of waves since. 3t/2. are presented in Fig. Very useful nomograms for determination of phase and group velocity of Love waves for a wide variety of cases have also been prepared by Sato [147. (4-205). condition the wavelength in the lowest mode becomes with H. as c approaches /Ja. . This difference in behavior results from the phase change upon kHji reflection at the free surface. 4-2). Kanai's curves appear in Fig. Again. Values for phase and group velocity have been computed by Jeffreys [76].. (4-212) This period equation 4-2.81. M2 C7^2 ^^' . the period approaches a finite limit at the upper limit of phase velocity. Curves for easy determination of period and velocity for the Airy phase have been given by Sato [140]. tt. It is interesting to compare this problem with that layers (Sec. Wilson's data are presented in Table 4-2 and Fig. when which is zero for the Love waves and c -^ . and in both infinite of in the layer and in the substratum. Similarly. In both cases a single cases the phase velocity ranges between the wave type is wave velocities compared two liquid involved. the free surface here always an antinode for horizontal displacements. . Wilson [209]. This is one of the few cases where it is more convenient to obtain values of group velocity from explicit an expression than by numerical differentiation of the phase velocity. among others. 25 Ml 1 .0 ^ = 2.__ ili = io Ml 0.0 h / /^=1.5 / P2_^ Pi 2 2.0 -^ ( y 2 / /^ = 2 Pi 1.0 1. = 2 1. {After and group-velocity (f7//3i) curves of Love waves Kanai. 1.00 '^_.8 = 4.rr-^ "-.8 ( ^ = 27.0 Mi_.0 y^=1 Pi 2.1 "' 1. M.-0.0 1.I. 1 0.A LAYERED HALF SPACE 1.0 1.5 D 2 4 6 4 6 2 4 6 8 2.0 ^ / 1.I. 1.2 1.0 /^ 1 Pi ifl 1.5 • ^ 6 8 10 ^/ 1.-'' 1 .95 ( no D 2. 2 4 6 C 2 4 6 8 2.0 - 3.2 / 2. 1 1 .5 I.10 - 21 1.8 n R C) 1 1 .6 1.0 -. Phaseratios of mj/mi Z 4 (c/|3i) b O U 4 8 12 for various and P2/P1.6 - y^ 2.0 1 .0 L / 1.'i=i ^=1.5 M.1 0. with changed notation. 1 Pi i^=5o 1. ^=25 Mi -___ .) . U Fig. 11 1.8 D 0.0 1.2 1.4 1. ^.0 _m£ ''-^=1. 1 .2 4 6 1.8 1 0. 1 .0 Ml ^ -^'^ ^^^-^^ ~~ 1 .2 2.0 1.. 4-51.2 1.4 1.4 H 6 2 3 4 2 4 6 3 1. ^^. r .4 1.0 iii = 2 5 Ml y/ 1 .05 - V\.0 h 2 4 6 3.0 0.5 1.1 1.4 J - Ml ^ ( 2 ( 3 2 4 6 D 2 4 8 10 1. ID «0 '-' "-tf t. 5? (B "*^ ^ &q •08s/'Ui>| ui A\\oo[a/\ dnoJO 212 . \ 1. Difficulties similar to those described for Rayleigh waves have delayed a complete understanding of Love-wave dis- .96 V^y \ \ ^..159. / / 10 X^ ^^ -:a ^ .08 "f \ node k \K\ d m 1.and group-velocity curves for case /32//3i = 1. A > \ \ \ lii \ \ 1 u \ \ 1. Therefore at large distances the latter dominate. Nakano [99a and 6] considered two kinds of solutions representing Love waves. the displacements at each point have both radial and transverse components (and no vertical component). first- and second-mode Love waves for source was discussed by Sezawa [167] and further developed by Sato [145]. The methods used are not unhke those given in Sec.04 \ \ s.28 V \ \ ^s s ^ V \ ] 1.. 4-53.16 V-\'-^ \ \ 1 c /3. 4-2.12 \ V \ \ \ 1 \ \ 1. will pre- Love Waves.5 0.24 \ 1. *« S^ \ 1. 0.297 and tiilin = 2. If it is assumed that the equal displacements are repeated in n sectors of the free surface.A LAYERED HALF SPACE 213 ::::::- 1.00 \ 0.92 0.1 50- 100 kH Fig.20 \ /3. across Continents.. In the case of axial symmetry the motion obtained does not depend on the azimuth and is entirely transverse. Phase. . Nakano proved that the amplitude of the radial component must diminish more rapidly than that of the transverse as the product /cr increases. 71 1.50 0.31 4.86 10.60 ft.60 c.66 17.10 4.47 0.72 2. km/sec 4.214 Table ELASTIC WA\^S IN LAYERED MEDIA 4-2.00 4.4 23.19 17.38 3.90 4.51 3.30 4.48 3.40 4.49 0.50 4.44 3.70 3.60 persion for continents.00 4. Numerical calculation of the dispersion curve by Eq.00 4. km/sec 2Tr/kH 1.02 5.56 3.40 4.80 3.70 4.7 1.00 4. km/sec 3.00 3.24 1.10 4.70 3.40 4.97 3.20 4.00 1.71 4. Jeffreys [76] gave the theory for the effect of a uniform increase in the velocity of shear waves in the substratum.00 1.00 4.60 3. In his most recent paper [197] on the subject.20 4. Stoneley made a computations for a double surface layer. km/sec 3.60 3.50 4.45 4.10 4.11 7.27 6.50 4.80 3.24 4.79 4.84 4.50 3.795 3.48 3.01 5.37 10.50 3.88 3.9 0.44 4. Theoretical Dispersion of Love Waves M2/M1 /3i.20 4.40 4. and the many attempts at fitting the observations on the assumptions of this simple type of structure have failed when extended to the entire range of observed periods.43 3.02 16.00 4.00 1.03 7.00 1. (4-212) for a single homogeneous layer over a homogeneous substratum is relatively simple. series of investigations involving .17 1.48 U.30 4.72 2.59 3.50 3.57 3.50 3.23 4.67 4.97 4. In an effort to obtain better agreement with Love-wave observations and to take cognizance of the layering deduced from near-earthquake studies.36 4.22 4.80 4.96 3. he offers only tentative support for his preferred scheme of layering and points out that the need for further work is manifest.70 3.40 4.55 5.90 4. A dispersion curve covering the period range 8 to 140 sec was Fig.. 4-52. The older method is to determine period and travel time for the first readable wave or for a conspicuous wave in the train and to construct the dispersion curve by obtaining one or two points from each of many seismograms. The preferred method now involves analysis of the entire train of waves on each seismogram. It is usually unnecessary to derive true ground displacements from the seismograms prior to dispersion analysis. N. 7-3). Palisades of July 6. using only seismograms which provide a clear train of waves in which the period varies gradually with time. 4-54.1 km/sec for periods greater than 30 sec.1 km sec for most of the period range 20 to 140 sec. This confirms the seismic-refraction determination of . Y. By using the method on seismograms from suitably selected earthquakes and propagation paths. this theoretical curve would fit the observed data to about 0. New sources of data from explosion seismology concerning the structure of the upper laycirs justify reconsideration of this problem. 4-54). the limitations of all seismographs used must be respected in order to avoid errors from instrumental phase shifts. NS seismogram (Fig. It may be seen that the effect of the gradient is to lower the group velocity by about 0. However. Wilson and Baykal [210] have discussed the two general methods which have been used in reading data for surface-wave dispersion from seismograms. These curves are for similar structures except that in one case the mantle velocity gradient is allowed for by Jeffreys' method obtained and is in the mantle. To show the effect of heterogeneity two theoretical curves taken from Wilson have been plotted as dashed lines.A LAYERED HALF SPACE 215 Wilson [209] used this theory in his study of Love-wave dispersion but concluded that the velocity gradient in the mantle could not he deduced from it. were recorded for a large range of periods on the Palisades. (Sec. 1954. 1954. NS seismograms showing Love waves from the Nevada earthquake plotted in Fig. Love waves from the Nevada earthquake of July 6. If allowance is made for the effect of the mantle gradient. the consistency of the experimental data may be greatly increased. 4-55.5 km/sec for periods greater than 20 sec. For the thin superficial layer indicated by seismic-refraction results. the periods involved were too short. Beyond this angle. A decoupling effect would occur for angles of emergence cos"^ /3i/iSmax. A velocity gradient in the crust. cannot be obtained with the accuracy required. An alternative method of reconcihng the theoretical curve with the experimental data would be to assume a shear-wave velocity of about 4. 4-53. reflection. It is clearly necessary to investigate periods shorter than 20 sec to study such unexpectedly thin layers. the cutoff being at about 13 sec. respectively.2 velocity in the crust For periods less than 20 sec the observed points rapidly limit. as may be seen from the curve in Fig. particularly for periods greater than 20 sec.to 6-km depth commonly found in the refraction work. One involves refraction. The most recent studies of Love-wave dispersion across ocean basins by Caloi and Marcelli [16] and Wilson [209] reach conclusions which support the result found in seismic-refraction measurements that there is no significant thickness of granite under the ocean basins.5 km/sec in the substratum. This corresponds to prolongation of the wave beyond the theoretical an effect not fully understood. However. in general. No calculation is available in this case.4 to 4. this would conflict with observations from explosion and rock-burst seismology and must be rejected. However. to about 0.1 to 0. The other suggests that it Another possibility of explaining the low velocities in the period range 8 to 30 sec was eliminated by calculation of the second-mode curve shown in Fig.216 /Sg ELASTIC WAVES IN LAYEEED MEDIA km/sec and suggests that any vertical variation of must be small. the energy for the corresponding period (and all shorter periods) may be considered as confined to a "sound channel" bounded by the free surface. Since the oceanic dispersion is so small compared with the continental dispersion. the correction of the continental part of a mixed path is most important and. would strongly affect short-period Love-wave propagation. which involves reflections at the top and bottom of the layer at near-grazing angles. and scattering. Thus reflections from the interface do not occur. represents an effect of low-velocity sediments. the theoretical group-velocity curve has its minimum at a period between 2 and 4 sec and has essentially reached its constant limiting value of velocity of 4. and a parallel plane above the interface. where I3i and /3max are the shear-wave velocities at the top and bottom of the crust. all the investigators have deduced values for the depth of the Mohorovicic discontinuity which are several times greater than the 5. It seems that the high values of layer thickness . as revealed by explosion and near- earthquake investigations. fall below the train theoretical curve. Two explana- tions for the behavior of the short-period waves have been advanced. Love Waves across Oceans. Although the velocities were considerably lowered. _o u: T3 O (-1 03 a a o •09s/'UJ>i ui Aipoi9A dnojo 217 . 52 km/sec. This accounts for the brief duration of Love waves on dis- seismograms of earthquakes for which the path the curves for is principally oceanic. km are given in Fig.218 ELASTIC WAVES IN LAYERED MEDIA found by Wilson and Caloi can be attributed to the fact that they confined their attention to waves of period greater than 20 sec and to propagation paths which often included large continental segments. Ignoring this latter part of the surface-wave train. Ewing. we can extend the observational data on oceanic Love waves to the shorter periods required to determine the crustal structure. and Press [111] of the Honolulu seismograms. 4-55. a conclusion also For periods greater than 18 sec the observations fall about 0. He infers a crustal structure consisting of 2.87 km/sec. until better-developed short-period Love waves over additional Pacific paths have been examined.71 km/sec. the observed variations of velocity with period becomes small for periods greater than 20 sec. H = Q km. being so severely attenuated at continental boundaries that only seismographs on oceanic islands or extremely near continental boundaries can receive them. they are usually absent. [33] presents dispersion data for a Pacific path which agrees with our data for periods greater than 20 sec but gives significantly lower group velocities for shorter periods. Sedimentary jSz and oceanic layers need not be considered small or vanishing rigidity. It is clearly necessary to investigate periods shorter than 20 sec to 15 tinguish between H = 6 km is and H = km in Fig. = 3. and Evernden a mantle with velocity 4.1 km/sec below the theoretical curve. This discrepancy is of the proper order for the effect of the velocity gradient in the mantle dis[170]. for the neglect of the shorter-period Love waves. As expected for a thin superficial layer. The second reason is that at a period of about 8 sec the short-period Love waves merge with a prolonged train of oscillations which record on all three components with roughly equal amplitudes but without systematic phase relationship.76/ii. as was shown in a study by Oliver. Theoretical curves for the cases H2 /3i = 1. reached by Sezawa cussed in the previous section. Not all oceanic earthquakes excite the short periods. Since oceanic dispersion for Love waves small compared with the cordifficult to correct responding continental dispersion. this result. of oceanic crustal thickness it is important yet for the continental portion of the path.31 km/sec. 10 km with 3. . 4-55. for Love waves because of their The curve f or i^ = 6 km fits the data reasonably well.50 km/sec. and H = 15 = 4.5 km with shear velocity 2. We prefer to reserve judgment on which proposes a novel crustal structure incompatible with seismic-refraction results. Two papers present divergent interpretations of oceanic Love waves. We attribute the excessive values deduced in earlier investigations to the fact that few data were available for periods less than 20 sec and that propagation paths contained large continental segments. There are two reasons First. 4-55). a value essentially equal to the velocity waves in the upper part of the continental crust. As little as 2° intervening ocean is sufficient to eliminate the phase entirely (Press and Ewing [125]). Auckland. Ewing. on seismograms from Honolulu. short-period shear waves polarized vertically (SV) occur in the second and higher modes of Rayleigh waves propagating in the crustal layer (Fig. The Lg phase is a short-period (1 to 6 sec) largeamplitude arrival in which the motion is predominantly transverse (Figs. All velocities less than 4. If the partition of data is allowed. there will be a tendency for the short-period Lg phase . Coulomb classified his observations into two groups. Lg and Rg Waves. which is is in full accord with the seismic-refraction results.A LAYERED HALF SPACE 219 Coulomb [20] studied the Love waves from the Queen Charlotte Islands earthquake of Aug. Similarly. Lg is 3. velocity The wave with higher and may be considered to represent a different kind of phenomenon. If velocit}' gradients occur in the crust. as indicated in Fig.51 km/sec. Apia. The phase occurs only when the earthquake epicenter and the seismograph station are so situated as to make the path entirely continental.5 km/sec are taken to represent Love-wave propagation with the single dispersion curve indicated in Fig. Coulomb's data with that from Honolulu (Oliver. and Press [111]) and Bermuda (Ewing and Press. unpublished) give excellent definition of the oceanic Love-wave dispersion curve in the critical short-period range (see Fig. velocities related to the sinusoidal-wave trains often observed over continental mixed paths having (Caloi [15]). Short-period transverse (SH) waves propagating with this velocity are included in the classical Love-wave theory in the limit U = c = iSi. As an alternative explanation.85 km/sec. No model of crustal layering has been proposed which can transmit surface waves at velocities appreciably greater than 4. a difficulty that Coulomb pointed out in connection with his tentative suggestion that these were higher-mode Love waves. Both of these mechanisms The velocity of of shear can explain many of the characteristic features of Lg. 4-56 and 4-57) but accompanied by appreciable vertical components.05 km/sec to 34 sec and 4. The second group covers the range of periods and velocities from 8 sec and 4. Although the precise mechanism of Lg propagation is not understood. 1949. The first represents the ordinary long-period Love waves {G waves) in the period range 82 to 58 sec. between those of the phases SS and Sn It is puzzling that no single seismogram has been found which shows an unbroken train of Love waves covering the entire range of periods from 8 to 50 sec. it here suggested that Coulomb's data be grouped in another way. 4-53. 22. 4-35). 4-55. and Riverview and presented the first adequate data for the critical period range 8 to 20 sec. it is certain that transmission of shear waves through a very efficient wave guide is involved.5 km /sec. Palisades N-S 1 March 55 To = 15 14:02:25 Yukon Aftershock Tg -75 Fig. Palisades March 1.1 ^^^î i iyi/Ui -nFig. 4-57. magnitude 5. Lg phase from Southern California earthquake t-*-60sec. Palisades NS seismogram of of July 28. 1955. 4-56.— 220 ELASTIC WAVES IN LAYERED MEDIA Jl- Jy~Lg -h . 1950.3. NS seismogram of Lg and Rg waves from Yukon aftershock of . 1 minute- . The experimental procedure simply is to search for the phase on seismograms. It has been found without exception that the crust is typicalh' continental in any large area where the water depth is less than about 1. 221 such mechanisms are is its Two indicated in Fig. Channel along a velocity minimum Two possible mechanisms for Lg propagation. Lg propagation confined to the uppermost part of the layer. 4-58. Rg waves correspond to propagation according to the portion of the Rayleigh-wave dispersion minimum value of group velocity. which is A principal probUim of the L(j phase long probably caused by reflection and scattering rather than low values of group velocity. Channel along the surface fornned by velocity gradient Surface Mohorovicic disc. is with periods of 8 to 12 sec and is also restricted to continental paths. respectively. duration. In probability.SPACE to concentrate in the zone of lowest velocity. its presence or absence indicating either continental or oceanic plus continental path. Some recordings of Rg under particularly favorable conditions (see Fig. all 4-57) show inverse dispersion. In the second. 4-58. 4-36 falling to the left of the Lg waves may be used to determine . from their orbital motion and velocity. The phase occurs Lg waves Surface Mohorovicic disc.A LAYERED HALF . Lg waves propagate near the depth of minimum velocity. Fig.000 fathoms curve of Fig. whether the crust beneath a given area is continental or oceanic. Observations of Rg (scje Figs. 4-38 and 4-57) establish these waves as Rayleigh waves. In the first. C. Sezawa and Kanai [178] expressed the ampUtudes of Love waves in terms of the thickness of a surface layer.000 fathoms (Press and Ewing [125]. Q2. ^1 — ior h < < z < H Jn = -4e^"' Jo e""'" sinh v.222 ELASTIC WAVES IN LAYERED MEDIA and typically oceanic for water depths greater than about 2. apparently originating at the image source at 2 = —h. ^2 = = 4e"*" coskxdk (4-215) ''' ^2 4e^"' f Jo Sê'^"""'' s'mkxdk Similar formal solutions can be obtained for an initial S pulse. A conclusion reached by Lee [85. and Press [112]). D. He showed that for Rayleigh waves the effect of a thin layer is more pronounced for horizontal than for vertical motion. of a relation between the thickness of a layer and the amplitudes of waves propagating in the system has been discussed by several investigators. The sum of these is ^Q^ = ^Q -f <^^ = — 4e"" r" / dk e~''''sinhj'i/i cos/ca. If a line source is at a depth s = /i in a layer having the thickness H. Substituting these expressions in the six boundary conditions (4-191) and (4-192). the solution for an initial P wave is obtained by combining the direct wave (^ô) and the wave (v^r) reflected at the free surface. They discussed also (Sezawa and Kanai [179]) the analogous problem for Rayleigh waves.z cos kx — ^1 f or 2 < h (4-213) with the supplementary potentials ^^ <pi and xpi for the layer (0 < z < H) = = 4e'"' [ •^0 [Ae"'^""""" + + Be''-"""] cos kx dk (4-214) ^^ 4e'"' r Jo [Ce-''-'''''' De''-"""']smkxdk written in the form The potentials for the lower medium can be ^26"''"^^"' . As usual. Ewing. Using Sezawa's theory. the problem of two-dimensional propagation of body waves as well as surface waves in a two-layered sohd half space was studied in detail by Newlands [105] by a method different from those discussed in preceding sections. we obtain six linear equations to determine the coefficients A. Many examples of group-velocity curves for Love waves as well as for Rayleigh waves were computed by Kanai [81]. 86] should also be mentioned in this connection. each of these coefficients can be written as . S2. B. and Oliver. The important question concerning the existence Other Investigations. As mentioned above. The determinant equations A of this system of six may be written in the form where D^ are represented by expressions (4-198) given earher in this section. since each term of the series corresponds to a different kind of pulse. fc^2 • • • . all four branch points A-„i. z. etc. (see Sec. 2-5). Now. The integrals in (4-213) to (4-215) are of the form Jo G{k) cos kx dk or Jo kG{k) sin kx dk. If the complex where A. Each of these is generated by the branch line corresponding to Re Vj = 0. For (4-217) 2iri Ju ^ CO are given by (4-213) and (4-214). and must be replaced by the quotients mentioned above. must be considered since the separate terms are not necessarily even functions of Vj and v' (j = 1. Now if contour integration is applied to each term in the manner suggested by kffi. The main contribution to the integrand along a loop £„i. There are also contributions from the poles due to the existence of roots of the Rayleigh equation (2-28) and of the Stoneley equation (3-139). Lapwood (Chap. and H.. 2-5). the determinant A being written in the form (4-216). B. 2). This method yields an important interpretation of the expressions obtained. it can be easily seen that a general term of the series into which the cp and \p integrands are expanded contains an exponential exp [iut of the form — i^x — h^vi — h2v[] (4-218) where hi and h2 are linear forms in h. The contribution due to each term of the series representation of (4-213) to (4-215) is then composed of several parts. Newlands obtained their approximate values using the Bromwich [13] expansion method (see also Sec. A-^. <por. 25). The coefficients <Pi. as in Sees. the solutions (4-213) to (4-215) may be generalized for the case of an initial P pulse having the form of a Heavi< z < h side unit function. £si. 2. • • \f/i • variable f in (4-213) to (4-216) is taken instead of k. These expansions hold under the condition that the sum of the second and later terms in braces in (4-216) is very small compared with 1. Ref. or by a pole. 2-6 and 3-3.A LAYERED HALF SPACE 223 a quotient of two determinants. comes from the neighborhood of the corresponding branch point (fcai. The latter wave at the free surface which reaches the A second group of terms obtained by Newlands cannot be associated with similar paths. the branch t^ = = t---{H -h-^z)(K-Ky 0. 4-59). 4-6. - h(^. necessary to consider systems having more than one superficial important problem is that of Rayleigh-wave propagation along oceanic paths. One type of path associated with branch point ka^. It travels with the velocity aj in the substratum and emerges as a shear wave traveling generates a compressional detector at R. with travel time corresponding to the path SKMNR (Fig. They are called "blunt" pulses.224 ELASTIC WAVES IN LAYERED MEDIA • •). Since the corresponding pulse arrives at the instant t^ = 0. In many studies of surface-wave propagation layer. at the critical angle. . x.hy (4-219) When ^^ a pulse arrives. and the layer thickness H. The terms obtained could be divided into two groups. where a liquid layer and a basaltic layer of approximately it is An . where the variable t^ is a linear function of t. 4-59. this condition will determine the time required to travel from the source to the observer along a path. z. in the first medium with This represents a compressional wave propagating the velocity ai which strikes the second medium N 1 H «2 2 Fig. Three-layered Half Space. The first. For example. and appropriate values of 40 "zero-order" and "first-order" terms were computed by Newlands.and higherorder terms represent the effect due to the finite depth of the layer and to • the presence of the substratum. In the final form obtained by Newlands for a term of the first group we have a factor equal to the Heaviside unit function H{t^). Such a path minimum-time determined by the form of point ka2 leads to a term in which is t^. in the direction MN. In this case the coefficients vi and pi do not appear in the exponent of (4-218) but in the amplitude factor. The zero-order terms represent the generating pulse and those due to the free surface of the layer. A LAYERED HALF SPACE 225 equal thickness are underlain by the mantle.or Rayleigh-wave problems where velocity gradients in the crust or the mantle are approximated by several homogeneous layers.kH) sinh (n^^kH) + H' cosh {nJcH) cosh {yhkH)] X tanh {uokH) = signifies (4-225) water plus unconsolidated sediments. the condition that the determinant must vanish in order for a solution to exist gives the period equation. 4-5 accounts for the main features of oceanic Rayleighwave dispersion. These stipulate vanishing of stress at the free surface of the liquid layer. with thickness approximately equal to the depth t. according to many geological speculations about the Atlantic Ocean. + H. it has been necessary to make calculations on the effect an interposed solid layer. 1/^2 + Ee-ne--^ = {Me'-'' + Ne-"'']e-"'' = ^^-'^^-"' . continuity of normal stress and displacement and vanishing of tangential stress at the Hquid-solid interface.<e (4-223) .+//. lo + U suih (riikH) sinh {n^kH) + H sinh {n^kH) cosh (ii^kH) + U cosh (uikH) sinh (n^kH) + H cosh {7iikH) cosh (rhkH) + [Zrsinh {n.\De- ^^^ < z < H. and continuity of normal and tangential stress and displacement at the soHd-solid interface. we substitute expressions (4-220) to (4-224) into the boundary conditions to obtain eight simultaneous linear equations involving the As before. (^-221) (4-222) <P^ for//. between the liquid and the mantle. As usual..kH) sinh {n:. according to refraction measurements. The result is eight coefficients. "liquid" here . Two cases arise one in which the layer is assumed to be granite. Although the theory presented in Sec. tAs before. There are now eight boundary conditions. we assume that a train of plane waves in the given half space follows (Jardetzky ^1 may be represented as and Press [70]): = {Be"' + Ce-'"'}e-""' for < H. Other examples involving multiple layers arise in Love. z < H. and one in which it is taken to be basalt. (4-220) . Oceanic Rayleigh Waves with Layered Substratum.kH) + Z^ sinh (rhkH) cosh (n.kH) + W cosh (n. of liquid — To derive the period equation appropriate for this problem. Vt^^]G.G.^| 182 + TF (4-230) = ^Vl'§. I -74 li = [2. 4 (4-226) 4 2\2 4 - [ o 2 - T/2 «1 Fi -2 ) 1 P3 ^4 Tr4 «! 7i P3 T74 "l nans Fi ^+4— .X' (4-229) where x = ^vl4P2 P2 2(^-1) \M2 / F= Z = Tf F?î P2 + 2(^\M2 1) / = F.Y' G^ = n.riM.njV . - 4G2 and 7// _ _ _ — = Pi 8 P1P3 îy^4 T^8 Ol P2 1^1 nona Tr4 «! 1^ 1 ^ |Q| 7// ^2 p2 no T" ^^ Pi /S2 4 tra (4-227) W Iff t'i ^—Vtf.Vli-2(^-l) / P2 P2 P2 \M2 \M2 X- Ff^ /Sj = 2(^ - . P2 nona ^ no tt8 «i 1 08 ''^ Pi Pa _ — ~ 2 P2 P2 with Fi = Oil (4-228) Gr = XZ . = Z' - n^n.WY G.226 ELASTIC WAVES IN LAYERED MEDIA where 2\2 4 7/ U = ^ 1 ^ — ] K 1 — + 4 — nin4 K ^ Tj . .56 1. 1 v/'Tr2 2 /i2 2 nz — /.50 8. /.57 00 Case 2 1 4. Using notations similar to those in Sec.67 3.00 2. These cases 2 and 3 are experimentally indistinguishable from each other for the periods between 15 and 40 sec. and (3) z = 2 = 00 to 2 = 0.-.52 6.00 3.67 3.10 1.10 3. Love Waves. are available for three cases. (4-225) to (4-231). we write the corresponding Case 1 differs by the study of well-chosen -H period ecûation tan {kyoH) = 72/X2 t TiMi + TsMs (4-232) 72M2 TiMiTsMa where 7.98 4. 4-61). These waves can also occur in more complicated structures.J A LAYERED HALF SPACE 227 and Tr2 2 Wo Vi - VI n.68 5. calculated from Eqs.52 (3. 4-5. (2) 2 = io z = .00 2.57 oo Cases 1 2 3. A necessary condition for the existence of Love waves is the presence of one superficial solid layer.00 1.oo. the range covered by observational data.18 4. — /. as follows: Layer Case 1 1 a. = ^/l - ^ 72 = 1-^ /So 73 = or A IS^ 1 ^ Pa (4-233) Real roots of (4-232) occur when < c < /S. -y Fâ? i «2 s/^Kfa? a^ /3. 4-60.00 H.52 7.57 5.57 00 show that Group-velocity curves for these cases are presented in Fig.57 5. «3 (4-231 \ \ ^ - n4 ^2 = \ \ 1 - Numerical values. from these by an amount which might be detected seismograms (see Fig.00 5. Stoneley [192] gave the theory of a generalized type of Love wave for a three-layered solid medium extended from (1) to 2 = -H. km/sec 1. gm/cm' 1. km/sec p.90 8.90 1.68 5. km 2 5. 57 1_ \ km.0 ^ ^ \ II e-~ -c ase III \ \ 1.6 " / / 15 ^^ :r= bi \ Case I (/f= 5. .1 0. Group-velocity curves for a liquid superposed on two solid layers: cases 1 and 3 with B.4 1 1 II D Case II Ca iel' 2.0 1.2 \ Casei 0. 4-60. Group-velocity curves for a liquid superposed on two solid layers: cases 1..) 2.2 .57 km.2 • 4.8 o .6 ' 1. and 3. 2.7 km.) 3.8 \ T t^ cr- o- 0* [II Nr^ 10 0.4 ' Case I II (H= 5. 2.228 ELASTIC WAVES IN LAYERED MEDIA ^ 2.7 10 1 1 1 u- r 1 1 1 20 25 Period in 30 seconds 35 40 45 Fig.4 n 0.. 4-61. 5. = 5.5 50 100 kU Fig. 270 for case 3 71 also does. and M3 > M2 > Ml.second layer was then taken very large. on Love waves is greater than on Rayleigh waves. ^. three cases occur: (1) ^3 > c > ^2 > 0i. [198]. P3 > P2 > Pi. we obtain the corresponding period equations in which tanh -u-ill replace the corresponding tan function.y)^ = (Pzy)3 at 2 = H2. and (. increase with the depth.3) the subjacent material taken as ultrabasic rock. Some general conclusions maj^ be is drawn from these period equations. Love waves in two layers in contact. To determine crust. Mi7iM3T3 tan {ky.A LAYERED HALF SPACE 229 The problem and Tillotson each having of propagation of (three-layered solid half space) [194]. Stoneley [195]. The resulting five homogeneous linear equations with respect to the coefficients A. > c. from = H2 to z = 00 For this case ^3 > ^2 > îLove waves may also exist when (83 > > jSa. (8. and 72 are real. E can have solutions different from zero if their determinant • • • . and Stoneley Jeffreys [77] studied the formation of finite thickness.velocity layer In case (1) we put f. (2) 03 > 02 > c> I3„ and (3) 03 > ^2 > /5. B. gated by Stoneley The effect of the low.= V'-| (*-2^^' and (4-235) The boundary conditions such as (4-208) to (4-210) must be supplemented by the equations V2 = ^3 and (p. from z z = to = H2. In the limit the thickness of the . = ^/|-l = ^|-l 7. The low-velocity layer was found to have little influence on the surface amplitudes of Love waves in the range of periods customarily studied. Stoneley the structure of the continental parts of the earth's granite. from 2 . since the latter can exist in a homogeneous medium.This ca.) tan {ky2H2) = where 71 (4-236) we may observe that 72 becomes pure imaginary.H. under the condition that the velocities /?. The dispersion curve continuous . is zero. Putting 70 = and fi = iyt. Love waves in a double surface layer was considered by Guterjberg [o3]. For a high-velocity intermediate layer there is a certain critical wavelength beyond which the Love waves will not exist.se was investiz = to and Tillotson assumed the following layering: (1) —Hi. Now. This gives the period equation as follows: ^421^2 tan {kyiH^ — JU272M373 + M1T1M272 tan {ky-iH-) + To and pass to case 2. (2) an intermediate basaltic layer. the thickness of the sedimentary layer of the continents was determined as about 3 km. Period equations can be written to conform to the different cases which are determined by values of ^j and c. 413] (see Sec. Stoneley and Tillotson made numerical calculations for H2 = Hi. the group velocity can be readily found. and Stoneley [195] has investigated Since the phase velocity c is the case H2 = 2Hi. planes as in the two-layer case.230 through the value c ELASTIC WAVES IN LAYERED MEDIA = /Jz- For cases 1 and 2 there exist one or more nodal 3. No roots of Eq. and we obtain a vanishing determinant of the seventh order. 2-8) treatment of the effect of a disturbance produced by a traveling line source. (4-236) exist for case determined in terms of k by this equation. resonant coupling may occur for a particular freis slight. there can be a large discrepancy in the estimated thickness. In some cases. As a necessary condition. He concluded that one can find a dispersion curve corresponding to a single layer which fits that of a double-layer structure very well. so that even though the energy flux across the interface the intensity of the signal transferred structive interference. However. hoAvever. because of the great density contrast. Sato [143] also examined the question of using dispersion curves corresponding to an equivalent single layer instead of those for double superficial layers. The traveling be replaced by a succession of infinitesimal impulses placed at equal intervals of time along the path of the disturbance. the effect of the atmosphere may be neglected. he found that the velocity of distortional waves in one of the layers must be less than that in the semiinfinite substratum. 4-7. density. Air-coupled Rayleigh Waves. and constructive interference is possible only for those waves whose phase velocity c equals the speed of the traveling may . To a first approximation we may neglect the reaction of the surface air wave on the impulse and follow Lamb's [84. Sato [142] discussed the problem of propagation of Love waves in a double superficial layer with special emphasis on the condition of existence of these waves. Numerical examples are given for some cases. Each impulse initiates a train of dispersive waves. p. quency. The extension of these theoretical results to a triple surface layer resting on a uniform substratum was given by Stoneley [196]. and the range of existence of Love waves can be found. and rigidity of the layers. The number of boundary conditions is again increased. In most investigations of elasticwave propagation in solids in contact with the atmosphere. may be significant because of con- In the first case we shall discuss an impulsive disturbance generated in the air impinging on a plane interface bounding a system of solid or liquid layers in which free waves may travel parallel to the interface with a range of phase velocities including the speed of sound in air. As an application. Suitable Rayleigh-wave dispersion is introduced by the stratification of sedimentary layers. A problem of some practical importance is that of atmospheric coupling to Rayleigh waves generated in the earth by explosions. and use the notations and conditions given the preceding problems. is the group velocity corresponding disturbance to Co. The waves will extend ahead of the air pulse if U„ > c„. The potentials which satisfy boundary conditions can be written in the form (jffo = Ae''°'Jo{kr) for — <» < for for z < —h z (4-237) n = ^1 '/'I {Be'-' + -t- Ce~''°'}Jo{kr) -h < < (4-238) = = [De'-' Ee~'"'}Jo{kr) <z < H (4-239) (4-240) {Me"'' + Ne-"''}Joikr) lor <P2 H<z < • (4-241) oo ^^2 = Pe~''''J.p. To determine the nine coefficients A. 2 the densities and velocities of in the air and in the solid and 2.Me'''" + Ne-"'") - tiM' + k')Pe-'''" = (4-246) .u^')Fe-''" 2nAk\Me'''" Ee-"'') -{- Ne"''") + 2t. The preceding medium and the layered medium discussion also applies if the disturbance is in the layered can be given in the following form: Assume a point source in the air at a distance h j above the plane media all 1 2 = 0.k' - p... H be the layer thickness. and /3.A LAYERKI) HALF SPACE 231 Cq.' - Pico')(Z) i?) ^-E) {p{' + 2.o:'){De''" -1- + Ee—") - {2fi.{kr) (4-242) ..of{B + - C) + {2miA.Fe-'"'' + nM' + ^'){.{D {2î. B. The energy thus transferred will form a train of constantfrequency waves. A quantitative treatment for the case of a homogeneous surface layer over a homogeneous substratum detector in the air. and the atmospheric disturbance has the same characteristics as the constant-frequency train mentioned above.(De'"" - 2n^v. 1. the "ground roll" of seismic prospecting. where t/.k' + - + Jc'XM + N) (4-244) p.2P2k'Pe-"'" = (4-245) 2n. a. P. and denote by p. dilatational and shear waves Let in = 0..jL. otherwise they will lag behind. The duration of the wave train at any distance will be proportional to |l/co — l/t/o|.p{k'(M = N) = (4-243) 2v. we obtain the following system of equations from the boundary conditions: p. Here the primary disturVânce in the consists of a train of dispersive waves. = = XZ . — V 2\2 ) Gi WiWa .{D - E) .n.n.W' .232 vo(B ELASTIC WAVES IN LAYERED MEDIA - C) - v. W = 2{^ - \ \Mi / / G.Pe-'''" = = 2Z (4-249) (4-250) (4-251) Ae'"' voAe"'" - + voiCe-"''' - Be'"') The at 2 last two conditions [see arise because of the existence of the point source = —h Eqs. = Z'~ - = n. 3. We can be easily eliminated from the system of Eqs. where it Z represents the strength of the source. 26) and Lee [85. we introduce the notations of Love latter in the layers.n^n. (4-39) and (4-40)]. (Chap. From the last two equations follows that B=A--e-'°'' Vq C = -e''' Vq (4-252) These two equation. == ^1 l) / -^ = V'^Pr X = Ply-" Pi -^[^ -\\ \Mi / 7 = + 2(^ Vi r+ ^Mi Tf (4-256) z = Piy^ -Y'' -2{^ -\\ = pi X -Y^ n.k\M + AT) = = (4-247) + De"" k\Me'"''' + Ne-"''") = k^Pe'"''' (4-248) + Ee-''"" - Fe-''" + v[{Me'''" Be'-' Ne^''") Ce"""" + v!.Y' G. which were used by the investigation of propagation of Rayleigh waves in solid problem has been discussed in Sec. the frequency equation takes the form (Jardetzky and Press [69]) shown in (4-253). coefficients (4-243) to (4-251). page 233. In order to develop this determinant.X' (4-257) and = = 4(2 (2 - V')G. We now put This V = ^ ^1 (4-254) Ho = A 1 - \ 2 ni - A 1 ao \ 2 cCi n^ \ \ / 1 2 eta (4-255) n3 = Vl - r r n. restrict ourselves to the analysis of the frequency After certain transformations of the determinant of the preceding system of equations. Ref. iriinsGz ^ ^1^4 ttS P0P2 2 V Pi rions . 86].WY k Zi G. 4-5. i. O ?1 33 m 02. 3. 3 <£ 1 -M .A LAYERED HALF SPACE CO to 233 + 3 c^ 3 + 3 riS a. % (N . 1 (Mica tq o 3 tq 1 I 3 3 ^ iT ^ 1—1 1 c^ •m 1 a 1 (N r^ (M N r^ '- 0) £ I-H 3 Q. 4-202). 4-62).234 ELASTIC WAVES IN LAYERED MEDIA Pirii pi pi no (4-258) ^3 = -(2 . corresponding to a air. since c > a^ for this case. real values of kH must is correspond to the interval 0. the roots of the frequency equation become complex.001 ^ Ml = 13.3375. In the case of air-coupled Rayleigh waves. - ^"^r Wo Then the frequency equation takes lo the form sinh (riikH) cosh {nJiH) + + li sinh (jiikH) sinh (jizkH) ^3 cosh (njA'H) sinh (nsA. At this point.//) cosh {n^kH) po = (4-259) The last terms of the coefficients air.9194 < V < ao/i8i- The upper limit equal in this case to 1.vy ^ ^ Pi 7^ + ^ Pi Pl n^nsF^ + ^^2 J_ y^g^ W3 Pi noTis k = {2.3374 The real part of complex roots above the critical value is very close to the corresponding roots of the frequency equation without air-connected very little the critical point <V< . we note that for po = Eq. deviates to the left and intersects the V axis at a point 1.3375 (= 1. 4-62: tto = 1070 ft/sec ^1 = 800 ft/sec ^ Pi ^ Ml = ^ 3. For larger values of V.and group-velocity the influence of the curves of Fig. it 1. The phase velocity may be calculated from the frequency equation (4-259). -4G. Ij having the factors and 1/no represent As a check.vyo. The following data were used to compute the phase.77 ^= M2 1 These values were chosen as most representative often encountered in seismic prospecting. and the air-coupling term of the frequency equation which contains the factor Po/piô is very small. The differs phase-velocity curve in the case of air-coupled Rayleigh waves from the curve given by Lee until the neighborhood of V = 1.3375 is reached (Fig. It effects are negligible is of near-surface conditions obvious that air-coupling when 0. using Lee's computations radiation of energy from the ground to the to obtain the first approximation. The roots of Eq.9194/3i > ao. Group velocity U was obtained by graphical differentiation of the phase-velocity curve.39 Bi Pi = 0. and the group velocity by graphical differentiation.14 Bl Pi = 1. (4-259) were computed by successive approximations. (4-259) for case single Lee the of a surface that given by layer reduces to [85] (Eq.070/800)./^) + + ^2 ^4 cosh (njA. A LAYERED HALF SPACE 2. each of which represents a different train of waves.3375. increases as the group velocity decreases.2 yi 1 jA 0. Branch I also accounts for the dispersive Rayleigh waves usually associated with ground roll.8 \ lîii "^ —__ -' \ 0. 4-62.2515 of the line V = 1. I. The group-velocity curve in Fig.4 \ \ \ 2. arrivals traveling with the speed of Rayleigh These waves first appear as long-period waves in the bottom lajêr. 4-62 divided into branches III.4 \ \ An k 4 6 kH Fig.9194/32. Succeeding waves gradually decrease in period. and the results are similar to those obtained by Lee. In Fig.8 V 235 \ 2. 4-62 the heavy lines indicate where atmospheric influence is negligible. II. But it deviates to the right in the neighborhood of the critical value and "ends" at the point kH = 4. Phase- and group-velocity curves for air-coupled Rayleigh waves. Branch I corresponds to the dispersive train of Rayleigh waves observed on seismograms of earthquakes. terms.0 \ \ 1. U = . Waves with group-velocit}^ values near the 0. since kH Waves continue to arrive with decreasing period vmtil a time corresponding to propagation at the minimum value of group velocity.6 '\ \ "] 1. The dashed lines represent new branches introduced by is air coupling. These waves are attenuated. Press The introduction of air shooting in seismic roll generated by air shots and Ewing [122] reported results of a series of field experiments in which the elevation of the shot position was varied from 30 ft above the ground to 40 ft in the ground. In these seismograms the first three traces represent reception at 2. There is an additional branch with dispersion features similar to branch III. the ground roll from the buried shots (Fig. In marked contrast. 6-3. we might expect these waves to be prominent for a source in the air recorded by a pickup on the ground and for a source within the ground recorded by a microphone in the air.650 feet.44/3i. It is interesting to note that the character of the ground roll from an air shot is independent of shot point elevation in the range to 30 ft at horizontal distances which are large in comparison with the shot elevation. This train begins at a time corresponding to propagation at the speed of sound in air and continues with almost constant frequency until the time t = r/0. 4-62. Branch II represents a dispersive train of waves beginning as a highfrequency arrival at a time corresponding to propagation with the velocity of Rayleigh waves in the surface layer. Retrograde elliptical particle motion shown by the first two traces proves these to be Rayleigh waves. Ground roll consists essentially of a constant-frequency train of .200 to 2.200 feet (1) from a radial horizontal geophone. (2) spurious and (3) from prospecting has stimulated studies of ground a vertical geophone.236 ELASTIC WAV'ES IN LAYERED MEDIA minimum in branch I have phase velocities approaching the speed of sound in air from the side c > ao. The succeeding traces are vertical geophones spaced on the air-shot record waves immediately following the air wave. The phase velocity of these waves remains close to the speed of sound in air for about 6 cycles. 4-63) 50 ft apart out to a distance of 2. The phase velocity of these waves should be close to the speed of sound in air. Branch III represents an additional train introduced by coupling of Rayleigh waves to atmospheric compressional waves.650 ft. The seism ograms shown in Fig. 4-5). From the qualitative discussion at the beginning of this section and from the discussion of air-coupled flexural waves in Sec. corresponding to the complex phase velocities whose real parts are greater than but close to ao- Higher modes of propagation exist at correspondingly higher frequencies. when the waves of branch I and branch II merge to form a single train of waves having a discrete frequency. 4-63 cover the horizontal distance range from 2. since kH is complex and has an increasingly large imaginary component as c —> aoOnly real parts of kH have been plotted in Fig. Air-coupled Ground Roll. The frequency of these waves decreases as time progresses until the time corresponding to propagation at the minimum group velocity. (see also Sec. These statements could be established rigorously by a very tedious calculation of amplitude functions. 14 0.20 0. With the use of a Fourier was obtained as a function of frequency for these waves (Fig.16 Period in shot 800 0.A LAYERED HALF SPACE 237 Fig. analyzer.26 seconds Fig. B. phase velocity the theory developed in the preceding pages.22 0. 4-63.12 0. The characteristics of the ground roll for air shots and buried shots are thus seen to confirm consists of a dispersed train of Rayleigh waves. 4-64.24 0. 1800 1600 ^ / c^-^ -^ o <u .) by Fourier .10 0. Dohrin. It was also found that a microphone situated a few feet above the ground detected a train of constant-frequency waves from a buried shot. It was found that the frequency of waves whose phase velocity equaled the speed of sound in air was identical with the frequency of the waves following the air pulse on the air-shot records. 4-64).E 1400 o o > 1200 1000 / /K 0. Phase-velocity curve for hole shot and air shot as determined analysis of two records at 800 ft and 1.18 0.200 ft. {Courtesy of M. Contrast between dispersive Rayleigh waves from a hole shot at 80-ft depth (upper seismogram) and constant-frequency Rayleigh waves from an air shot 10 ft above the ground (lower seismogram). At the free surface the half space are represented discussion of boundary conditions for the n-layered elastic by the first set of Eqs. have to be written separately for each homogeneous part of a medium and the appropriate values given to the elastic constants and the density. the terms being Fourier integrals. The boundary The conditions can be formulated in different ways. (4-4). 183]. The transmission of plane compressional waves through a system of alternate layers of two different substances. air. Therefore this discussion is usually reduced to some special points. The complete the problem of wave propagation in an n-layered medium presents very great mathematical difficulties. acoustic filtering. A derivation of period equations based on the condition of constructive interference is given by Tolstoy and Usdin [203]. by using a the wave equation. Thus the principal aim of Rayleigh's investigation [130] layer of a stratified was to determine the reflection coefficient in the first is medium which composed of a set of equal parallel homogeneous In optics a similar problem has been discussed by several writers (see. equations of motion such as Eqs. (4-1). Haskell [62] reformulated the problem in terms of matrices and suggested a new systematic computational procedure. we have conditions (4-2) and the second set of Eqs. The general equations which hold for a stratified medium were given in Sec.238 ELASTIC WAVES IN LAYERED MEDIA Benioff. We have seen the importance of the period equation in every particular problem shown before. For each . 165. Assuming a welded contact between the layers. The integrands in these expressions were determined as usual by means of boundary conditions. Forsterhng [47] and van Cittert [206]). for example. Brekhovskikh [8] suggested a new method of deriving the reflection coefficients certain differential equation of the first order instead of Using the formulas developed by Thomson [202]. By contour integration these integrals could be evaluated and some general conclusions drawn. 4-1. Remarks concerning the Problem of an w-layered Half Space. was investigated by Lindsay [88]. for example. Ewing. in connection with the question of layers. (4-4). Some general properties of solutions for n parallel layers in a half space will now be discussed (see Jardetzky [71]). They first considered a single layer embedded in a medium. and Press [4] have suggested that the low-frequency sounds often reported as accompanying earthquakes illustrate the same phenomenon and are principally confined to regions in which the superficial strata can propagate surface waves at velocities near the speed of sound in 4-8. obtaining a solution in series form. written about other particular cases of the The transmission of compressional and distortional waves through layered media in a direction normal to the boundaries was investigated by Sezawa and Nishimura [159] and Sezawa and Kanai [164. Numerous papers have been problem. k pjOi 2jXjVi — (4-265) substituting (4-260) in (4-262) to (4-264) each boundary condition be reduced to the vanishing of an integral taken with respect to the parameter A.ikr)e" dk yPi [ Jo 0.(t + dr 2M.A LAYKKED HALF SPACE layer the potentials ^.Q[ + 2y. They can be found as usual by solving the system of linear equations determined by the boundary conditions.(kr)e"''' dk -\- \ QYJ.VV Gj + = 2m.Q[' + n.. ^= — + = 2m. S'.-^.{kr)e"'' dk with Qn sional ^ S^' is waves situated For the layer in which the point source of compreswe must add the potential cpo = re-^'-'^Joikr)^^ Jo Vj (4-261) There are 47i — 2 unknown coefficients Q and S in Eqs.^ + a.{v'^ + ].v. are <Pi = = = f Q:J. and (p.{kr)e-'''' dk + ( J is SYJ.S'.A.r)^ = X. -e""" (4-267) Likewise. for the first interface at z = H^ we obtain the four equations: ^-viSê-^'"^ - v'.v.J. Wi.'e'^'"' = _^g--<^-" (4-268) . (P.^)i.. 239 and \pi. like those in Eqs.^ ^ Xjkai • (4-264) where 2n. ^r VdWj\ dr / + _ ^) d '"' = ^ + *^' M. we must use the expressions for (li. (4-11) to (4-14).')S[ + ixM' + k')SV + a. we obtain for the free surface the z two equations for the vanishing of stress components at = 0: -2ix. In order to write these conditions.(^r W-262) = .kMSi + 2fji. Taking the source to be in the first layer at a depth h..- + /:^)V'. (4-260).kMSi' = -a.- (4-263) (p.r)i: « = (VJ. A sufficient condition for this is that the integrands vanish. we must use for this Upon may layer the potential <Pi = (pi -\- <po- Equating the integrands to zero and canceling common factors such as the Bessel functions.Q'/ = 2fx^ke-'"' (4-266) a^Qi - 2n. Q'.::ie"'"-'^"-' + 2ix^^.k\[S['e'"'"' - aMe'"'"' - a^Q'^'e'"' + 2î.kV^.e''^-'— . • • - .k^'. respectively.Qie-''"' 2fji. For the last interface we finally have - ae"""^""' + - v'r.e-''''—' + Mn-i(^f-i + k')S'J^.Srê-''"' —= = (4-272)t + + + k'^S'^L.kY^.k\^S'.e"' -''"-' + vMne'""'"-' k'Sê'""''-' (4-273) -2Mn-i»'„-iQn-ie"'""'""" M„_:(.240 ELASTIC WAVES IN LAYERED MEDIA - v.e''-''-^ + /c')^„e""'"'^"-' = 2M„j'„Q„e"''"^""' a„_i(?:_ie"''"-^^"-' + a„_iQ.ar^Qrê"""" '' + 2n^kYSr. which will from the above only in the absence of terms for the source potential and in the subscripts.'e"'"' - k'Sê-"'''' - k'S^V'"' = ke""'"'-''^ (4-269) - H2W..'e''"^ = a.V''"' (4-271) = Similar equations differ -a.-.ke-'''"'-''' (4-270) + a.e-'''-''-' .e-'"''"'-' = Univ'n (4-274) (4-275) Thus the boundary conditions lead to a system of 4n — 2 linear The determinant of this system is (4-276).S'.ixM + k')&'. = ^ for Si-^ = n).2. g (4-277) where / denotes subscript g or s ' or " (can be omitted for g = 1.2yi„_.L. Q„ and Sn- taken for fThe index prime has been omitted . (4-266) to (4-275) are Qi equations.:'_i + 2M„-iJ'„-iQ:iie^''~''""^ + k')S'r. pages 242 and 243.-e-'^'"'-'' may be written for the other interfaces. and A is given by (4-276).S'^-.S'r. Then the solutions of Eqs.n.' + k')Sê-'''"' .k\[S[e-'"'"' + 2n. is and the a coefficient Q shows that the corresponding determinant or S.e-'''"' - 2n.QiV'"' - 2n. . separately. e.L>M = Z' (. = X=l Z (-D'^XDx^. o- + ' -|- 1 in the last because the index in the first • • • . as may be seen directly in (4-276). As we have seen before. However. exp ( — ViZ) and of /. that of a liquid layer overlying a solid half space.) depends upon the determined by the roots of this equation. 2. and q and Rx will replace a column with an odd number for f denoting the index and a column with an even number for the index ". Then (4-280) K. the sums of the corresponding terms. let us form the sums in brackets.... Scherraann's long proof [151] of the existence of roots shows the difficulties of such a discussion. to other considerations concerning of each variable the determinant A and 1. The period equation (4-278) is too complicated to permit a useful general discussion of its roots. This determinant of the 4n Vj or I'-. therefore.. We restrict ourselves. A Now we is neither an odd nor an even function of can write this determinant in the form Vn and v'^.. q. In order to see whether this expression is an odd or an even function of v. w — 2 order is an odd function — 1. . for example. R.l)''X2£'xa = X=l • • • (4-279) where to D^. In contrast.) where a denotes a column number corresponding to a given combination g. are subdeterminants.A LAYERED HALF SPACE 241 Now the period equation A = expresses the fact that the phase velocity (c (4-278) co/A. therefore.g. these roots determine the poles of the integrands in (4-260). We the shall consider. the solutions (4-260) and (4-261). for j = 2. A = Z' (X=l 1)'''4. • • • . ' exp (Piz). n — 1. We obtain the determinants A^^ and A^. of Eqs: (4-266) g..[{-lf^'DU.e'''] (4-281) term is obviously increased by 1 term was changed to " in the last one. An important characteristic of the expressions considered above is that we have pairs of coefficients Q or S multiplied by exponential functions in which the exponents differ in sign.. sum = \T. even in one = frequency or wave length in a manner which is of the simplest cases. ^ = 1.e-^'' The exponent X + {-lY^'"-'Di[. in (4-277) by substituting the right-side members {R^) (4-275) in a column determined by the subscripts /. r -^ 3. o* 3.:i (M N^ .C 1 n'^ i -^ + ^„ ~ii 3.-^ IW ^ + VN =} (M ^ + o» * . _ ft: "^ :k o- 3 (M ri « % «a rT o> ^ 3.^ + ^. i -y i:^ 5. . -^ ^^ '^ 4) r- ^ + <N ? ""ic a. i:^ <ii . ^ 1 j^ 1 ^ + ^^r 1 1 ''^ 1 . (M •^ a o o =1. tQ + 3. . 03 + a.II ^ + CM o3. On considering wave propagation in a three-layered liquid half space. = Z'(-l)'"^^C'^x(„. \l/ . i. (4-282) where DU. we obtain the sum two functions + It is evident that gr Z d*C.244 ELASTIC WAVES IN LAYERED MEDIA (4-276) it is By easy to see that DU... noting that by (4-260) the potentials (p and \p depend on Vg because of the factors Q and S and that these factors are determined by are even functions of Vg..e-'''' Di[. but it holds also f or g = 1. We now are able to make a general conclusion that. (4-284) if there is a factor Vg present in the coefficient these elements itself... the first four columns of A and the next set of four we see that for g = I there are two additional lines but the elements in these lines display the same properties as all others with respect to their composition and signs. This final conclusion is n — 1. When evaluating the integrals (4-260) in the complex k plane it seems that each radical v requires the consideration of a branch line integral. = T. Either and the second The elements of the d'/ = cK = d. Thus by (4-281) we consider pairs of terms such as by Vg Now of inserting (4-283) and taking account of the signs.. The and the n 1.. g = • • • . (4-283) subdeterminants C'/ = C'^ does not depend on Vg. we see that (p and ••• — = numerators denominators in (4-277) 1. in all cases of wave propagation \}/ from a point source in a half space formed by parallel layers displaying ... are odd functions of these variables. an important conclusion From this property of the functions (p and about the existence of the branch line integrals can be drawn.e"' = Di[.e. Therefore. = DU... or d'J = -d'.. not in the exponent. determinant A (4-276) denoted by d^ and d'/ and belonging to two neighboring columns for Q or for S coefficients display certain properties..uUe~'"'"^'' 2 + e^"'^""^'K (4-285) both functions are odd in Pg.. Pekeris [116] found that two branch line integrals vanish.) set of Di[. 2.' (--^y^'^'dVaU. We will now denote d% or — Vg d%. 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J. 1931.. 1927. Bull. vol. Bull. 652-653. 173. : Akad. vol. Kanai: Reflections and Refractions of Seismic Waves in a Body: II. Earthquake Research Inst. 1-17. Bidl. Scholte. (Tokyo).. Wetenschap. pp. : Bull. of Shallow K. Sezawa.. vol. : and G. 380-387. Sezawa. 1932. Sezawa. 165. 1935. Scholte. 1-12. Bull. 115-143.: On the Propagation of Seismic Waves: III. Sezawa. J. Sezawa. vol. 1946. 160. K. vol. of Oscillations in L and M Phases. 1931. K. 1934. 13. Ned. pp. 157. Koninkl. Earthquake Research Inst. Proc. G. Stratified and K. 10-17. (Tokyo). 8. K. 1935. K.. Earthquake Research Inst. (Tokyo). 14. pp. pp. Ocean and the Continents. vol. 9. pp. 158. K. pp. 13. vol. Sezawa. 1952 (in vol. of Arbitrary Elastic and K.. vol. Sezawa. K.: Surface Waves in Water of Variable Depth. Layers. R. pp. B.R. 185. Kanai: The Formation I.. Soc. 1938. Kanai: Dispersive Rayleigh Waves of Positive or Negative Orbital Motion. in Reflected Seismic 188. Bull. Gerlaruls Beitr. 19. (Tokyo). and V. 184. pp. K. 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CHAPTER 5 THE EFFECTS OF GRAVITY, CURVATURE, AND VISCOSITY 5-1. Gravity Terms in General Equations. The problems discussed in the preceding chapters form the basic part of the theory of tion of a disturbance were simplified, wave propagation in stratified media. Despite the fact that the conditions of propagathese problems presented great it mathematical difficulties. in several cases, was possible to find solutions and the approximations obtained have proved adequate Nevertheless, for the explanation of many observed phenomena of wave propagation. which were not taken into account, may be of importance in some applications. In Eqs. (1-13) we omitted the body forces pX, Moreover, we considered wave propagation only in those cases where all boundaries and interfaces are parallel planes, but in some cases of impor- Three factors, . tance the interfaces are curved, usually being cylinders or spheres. third effect which has been neglected thus far is The that of viscosity or other deviations from ideal elasticity. We first consider the gravity terms in the equations of motion. Usually we can assume that there is a constant field of forces (X, Y, Z) acting. Since equations for a fluid must hold for small motions starting from an undisturbed state, we can consider the p initial conditions u = v = w = 0, = po, and p = po. It may be proved (Lamb [26, p. 556]) that in this case the velocity potential must satisfy the equation § = „V=, + (x|+y| + ^|) Gravity is (5-1) the principal force which concerns us in problems of wave propagation, and we can put X = Y = 0, Z = g in all cases where the boundaries between homogeneous layers form a set of horizontal parallel planes. It is, of course, assumed that the z axis is perpendicular to these planes and is taken as positive in the direction of the gravity acceleration. Thus, the velocity components of a of ip fluid medium being expressed in terms [Eqs. (1-14)], this function must satisfy the equation g = .V=, + ,|f 255 (5-2) 256 ELASTIC WAVES IN LAYERED MEDIA instead of the ordinary wave equation formed by the of a fluid the equations of first two terms. For the potential motion integral motion admit the | + y-C/ + where /f^ = FW (5-3) V = grad ip U= An potential of body is forces arbitrary function F{f) usually included in the potential (p. If the and fluid is incompressible, p po form takes the this equation of the velocity const, = = if we can neglect the square p = - Po -^ -\- PoU -\- const (5-4) We can now write p and take that value = ~Po-^ + Pog(z + const) (5-5) of the arbitrary constant which corresponds to the position of the origin of coordinates. As to the equations of motion for solid media, they also will be changed by the addition of a term representing the body forces. We write only the first equation of (1-13): P ^= = (X + m) ^ + mVw + + curl ^(\pi, pZ (5-6) To is derive the represented by a wave equations (1-22), we assumed that the displacement sum of two vectors: s(u,v, w) gradtp yp2, h) (5-7) <p and \^ being displacement potentials [Eqs. (1-20)]. We can write, in general, a similar condition for body forces: F(Z, Y, Z) = grad U+ curl L(L,, L^, L,) (5-8) Then instead of Eqs. (1-22) we obtain the equations § - aW =U ot - ^W. ^^ ot = L, (5-9) under the assumption that p is constant. Equations (5-9) have particular solutions (Love [30, p. 304]): (5-10) *• = 4^ /// M 4' - f) ''^' "y' "' THE EFFECTS OF GRAVITY, CURVATURE, AND VISCOSITY If 257 we again consid(!i" gravitational forces only, condition (o-H) reduces to Z = gz -^ const. Differentiating Eq. (5-6) the displacements v and w with respect to (X x, y, z and two similar equations for and adding, we obtain P If Poisson's H= + 2y)V"B + PV'C/ f/, (5-llj equation is used for the potential Eq. (5-11) can be expressed in terms of Q alone. To last term evaluate the correction due to fluctuations of the body forces the in Eq. (5-6) will be written in the form \dx in Eq. (5-11) dx I state. (5-12) where Uo corresponds to a certain undisturbed becomes p(V'C/ Then the last term - V'f/o) (5-13) By Poisson's equation we have, therefore, Uo) ^\U - = -47r/(p - po) = 4x/p^ is (5-14) set equal approximately, where / is the constant of gravitation and 6 to — (p — Po)/p- Then the equation for 9 will have the form P H= (X + 2^)V'e + iTfp'e (5-15) Jeffreys [19] solved this equation to the gravity term is insignificant for compressional and showed that the correction due waves in the earth. Since the curl of the gravity force vanishes, this force does not affect the propagation of S waves determined by the functions ipi. 5-2. Effect of Gravity of on Surface Waves. Gravity terms in the equations motion (5-2) and (5-6) produce modifications in the solutions for surfaceIn an early paper Brom- wave propagation. Rayleigh Waves: Incompressible Half Space. considered the effect of gravity on Rayleigh waves in a soUd half space. This effect was introduced in the boundary conditions, omitting [3] wich the mass terms in the equations of motion (5-6). Moreover, to simplify the problem, Bromwich considered an incompressible sohd for which 00 as —> in such a manner that \d = U remains finite. For the two-dimensional case, Eq. (5-6) and the corresponding equation for w take the form X —^ d\ an 2 d^w an , _ ,. _, 258 ELASTIC WAVES IN LAYERED MEDIA We also have 6 = 1^ dx + 1^ = dz (5-17) Assume u, w, and 11 to be proportional to exp [{(ut — kx)]; then Eqs. (5-16) and (5-17) become (V^ + kl)u = -fi ^ ox (V^ + kl)w = -- n oz ^ < 2 V^^n = (5-18) In order to satisfy the last equation, put n = where nklPe''""'-'-" for < CO (5-19) P is an arbitrary constant. We can define u and w by nkff dx (5-20) w = which will satisfy 1 ^2 nkff -dz all + „ Hat-kx)~y' Be 2 Eqs. (5-18), provided that the second terms in (5-20) satisfy the conditions (V' or if + kl)u = v" (V' + kl)w = (5-21) = e - kl (5-22) The requirement 6 = applied to (5-20) leads to ikA + /5 = (5-23) of is equal to the sum Now, assuming that the normal stress at 2 = elem.ent of height per unit area of an the weight by and (1-11), p„, we write the boundary conditions in the form w, n + 2.f+.p„ = M(| + t)=0 at. = (5-24) Substituting (5-19) and (5-20) in (5-24) and taking the determinant of the two equations (5-24) and Eq. (5-23), we obtain the period equation We obtain the reversed sign as compared with the Bromwich period equation since we make use of waves propagating in the positive x direction. To study the effect of gravity on surface waves in a compressible sohd THE EFFECTS OF GRAVITY, CURVATURE, AND VISCOSITY half space, J4 2o9 Love [29] derived the period equarion klk: kl 9 _ ^^1. h' + I l£^ rL ^' + ' k ; [k l{k kf, kl- k\ k likt, - ki)k' kl-^ k kt - k:j = (.5-26) where, again. V = v/c' — ka Ve - kl k. k (5-27) In deriving this equation the square of gk/w^ was neglected. If gk/co^ is taken to vanish, Eq. (5-26) is readily seen to reduce to Rayleigh's equation for the velocity of surface waves in a solid half space. derived For an incompressible body, /c„ = 0, and Eq. (5-26) takes the form (5-25) by Bromwich, provided that {v /kf is approximated by \{kMe To find 2)\ the effect of gravity to the form, vaHd for first-order it is convenient to transform Eq. (5-26) terms in the small quantity g/k^^, cAl where jS + (5-28) k^' Cr 5 = ^/jj./ p = velocity of Rayleigh waves in absence of = number which depends upon ratio ju/X gravity Table 5-1 gives approximate values for b and n/\ and Poisson's ratio a (Love [29, p. 160]). Table 5-1. c]i/0^ for several values of Effect of Gravity on Rayleigh Waves <T m/x 4//3^ 5 1/2 1/3 1/4 1/5 1/2 1 3/2 0.9126 0.8696 0.8453 0.8299 0.1089 0.0462 -0.0309 It is seen that f or o- = j, i, 5 = 0, and that 5 ^ f or o- ^ \. Since for most crystalhne rocks o- > we may conclude that the velocity of Rayleigh waves find, is, on the whole, likely to be increased by gravity, the increment o- being proportional to the wavelength. For the case for example, that the velocity of = ^, /3 = 4 km 'sec, we Rayleigh waves is increased by about 0.2 per cent when the wavelength is about 500 km. 260 ELASTIC WAVES IN LAYERED MEDIA In Gravitating and Compressible Liquid Layer over Solid Half Space. an attempt to find a theory for the generation of microseisms (Sec. 4-4), Scholte [49] considered this problem in two dimensions. For the velocitypotential ^1 in the water layer, Eq. (5-2) now takes the form (5-29) For a train of plane waves propagating (fl in the water we can put (5-30) = 6 By (5-29) and (5-30) we obtain u.= = «;(!>" + «:") + ivg (5-31) or .= -«ipi 2a? ^^ ^g'"^) (5-32) and — exp + ^ LO-l 2 t(coi kx) {Ae- .,. + (5-33) The velocity components of a water particle are dUi Wi W^ = dWi ~dt (5-34) or by Eq. (5-33) Wi = ^-- dx = —IK exp i{<jit — kx) {Ae-'" 2ai_ + Be'") (5-35) Wi dz — exp [ I iiut — kx) — r-2 [('" + 2f?K'" ico, -(»•'- 2f?)H [see Now we can write ipi = (pr where (pi is a displacement potential solid Eqs. (5-34)]. The gravity terms in the equations for the solid part of the system are omitted by Scholte. f For the displacement in the P2 ^72~ we have (5-36) = (A2 ot + M2J 1 OX d\p2 h M2V W2 M2 P2 where fit d(p2 _ d(p2 = — . dx was seen in the preceding dz dz h ^— dx is d\}/2 (5-37) pages that this approximation valid unless the waves are considerably longer than ordinary earthquake Rayleigh waves. THE EFFECTS OF GRAVITY, CURVATURE, AND VISCOSITY Potentials which vanish as 2; 261 — > 00 are taken as usual in the form iP-i = —e '/'2 = —e (5-38) where ?2 and v'2, must be negative imaginary numbers determined by the conditions vi = ki, - e V',' = ki - e (5-39) As usual, tta and 02 are the velocities of compressional waves in the second layer. Now we obtain, and distortional by Eqs. (5-38) and (5-37), ^^ (5-40) 01 The four coefficients A, B, C, D can be determined from the boundary is conditions. The first of these conditions all that the pressure is zero at the problems considered before we have assumed that this free surface is a plane (z = —H). If we also consider its deformation and denote by ibi the vertical displacement, we can easily see that the constant in Eq. (5-5) is equal to H, and on the deformed surface z = + Wi we have free surface of water. In —H -jt + By (5-33), (5-35), Wi 9A=0 (5-41) (5-41) = ibi-ico at z = —H, and we obtain A^-o:' + ivg + ^je-^ + bI^-co' - ng + = i^}'"'" (5-42) The other boundary conditions are the continuity of the vertical displacement (or velocity) and of the normal and tangential stresses at the interface 2 = 0. The first of these conditions is, by (5-35) and (5-40), -{iv + —^2) A + (ir, - ^jB = -iv,C - ikD (5-43) Now the tangential stress or, by (5-37) and (5-40), -2kv,C + (A>2 - 2k'')D = (5-44) 262 ELASTIC WAVES IN LAYERED MEDIA (5-5) By we have the stress on the side of water = and on the — PiAUco H h — 2m2 -T-^ oz (5-45) side of the sohd medium P^^ = ^262 + - gp2{w^z = (5-46) if, according to Scholte, the additional stress is put equal to the weight of (5-37), (5-38), j^ -iy^': the small column due to the deformation. Noting that X2^2 X2 = 92 («' - 2/3'), we obtain by jT^^ and (1-5) + 2k'\^ 2/ii 2kv 2 1^02 lP20i (5-47) dz Kp2' By (5-37) and (5-38), (w2)..o == --^C V2 k --D (5-48) Thus the boundary condition the form (2 for the normal stress may be written in = 0) p,A[io: + ^+^ 03 2aiico + _i_ R( PiB\ V loo — vg — + , g 2aiico cu = P2C ^coll-^) + ... 2fcM gv2 + 2ikj'2(jo gk (5-49) P2D L '^/32 Thus the boundary if conditions yield a system of four homogeneous equations (5-42), (5-43), (5-44), their and (5-49) which will have nontrivial solutions determinant A vanishes. Expanding the determinant, we find for the period equation (2s' - 1)^ - 4mm' - ^ 4(l CO S - ^•^^^'?^ + CO 77 -X.t^M 2ai 77 J 1, J + £i-t„m(l-(2|)V-2^^^ \C0 P2 [ = (5-50) / j T? where s = i'P2 ti'2 1% Cylindrical Curvature. the theory of wave propagation in a sphere will In the following sections be used to show the effect of spherical curvature. for a given period. as we might expect. only one type of propagation occurs. is omitted. Eq. and the solution is simply that for gravity waves on water. Eq. 5-3.„. giving the dispersion of Rayleigh waves in the system formed by a liquid layer underlain by a solid substratum (suboceanic Rayleigh waves). AND VISCOSITY 263 « 1 for naturally occurring waves. (5-50) can be approximated by (2s' - 1)' . If the axial component of motion p ^= 2M)i^-2 + -s. For short waves the solution reduces to that for flexural waves in a thin (Sec. It is useful to consider first the simpler of problem Rayleigh-wave propagation along the circumferential direction of a This problem was solved by Sezawa [50]. in Scholte pointed out that two different types of wave propagation are When c /Sz < "z. of magnitude of each other for actual conditions in Another example of superposition of two types of waves will be found problem of flexural waves in floating ice (E\\ang and Crary [11]).4mns - ^ — Xl CO - ^ OJ 1 fl s J I where vl = kl^ — k involved. For long waves the gravity term in the period equation predominates. Effect of Curvature on Surface Waves.^ . and the curves cross. Although both types of propagation are dispersive. ^ df ~ ^\ dr'' ^ r ^r da' . (5-51) takes the form of (4-154). The phase velocities for either system taken separately exist over period plate modified slightly ranges which overlap. Eqs. 6-3) in the by the presence of the water. Eq..THE EFFECTS OF Since S'//ca? GRAVITY. there is no wavelength for which the phase velocities « come within an order the ocean. . CURVATURE. (1-24) take the form (X -t- cylindrical surface. The separation into two types of propagation arises from the great disparity in phase velocities of gravity waves and Rayleigh waves. In this system. (5-51) reduces to that for gravity waves « an incompressible fluid underlain by an immovable bottom: 1 - ^ tanh kH = For g/k c^.+r^ dr a. ^{kJ)e''^'^'-" k„ r 12^ Displacements (5-54) qz and V2 consistent with the solution for given in and satisfying the condition ^ = are ''' ' (5-56) _ ^ 2j> dJ k„{k0r) dr i(al+kaa) kl Since the cylindrical surface are is free of traction.a{kar) dr (5-55) = -^J^J. q. k^ have the usual is definitions. (5-52) are d = A J Ukar)e' '"'*'"''' (5-54) where w. . where z are cylindrical coordinates. 5-4. fc„.V2 6. particular solutions of Eqs. period equation which differs in two terms from that given by Sezawaf {kl ~ 2kl)JUkaa) dJka{kaa) 4:k - 2 d^Jkaikga) da' Jka{kaa) {kl .2e)JUha) + I a T 2 dJUkea) da (5-58) a dJka{ksa) n = da t a da The subscripts ka are considered constant when the derivatives with respect to a are computed. I They are 1 d(rv) dv r dr r da 2a = _ Idq r da- (5-53) r dr Now. k. 12^ ment components The expressions respectively. = are Hut+kaa) ' ^' _ ~ A kl dJ^.264 ELASTIC WAVES IN LAYERED MEDIA r. v Displacements q and consistent with the solution for d given in (5-54) and satisfying the condition ^. and Jka the Bessel function. a is the radius of a circular cylinder. v Inserting in (5-57) the values for from (5-54) to (5-56) we obtain the . the boundary conditions \e + -\- dq 2/x dv ?. 1 dr q2 dr r r ^g da _ at r = o (5-57) where q V = = qi Vi -\. <t. Radial and azimuthal displaceare q and for 6 v. and can be obtained for cylindrical coordinates from the general formulas 1 d(rq) of Sec. obtained using model seismology techniques by Oliver tation of the corrected curve The compu- was carried out by Oliver. 5-1 for a It is seen that the theoretical soHd Avith a Poisson's constant of 0. In Sezawa's paper the corresponding derivation for Love waves is Spherical Curvature.5 Fig.2 7/3 1.THE EFFECTS OF GRAVITY. (5-58) 1.8 0.) = /x. and experimental points for a cylinder with Poisson's constant 0. controlled by elasticity. (5-58) shows through the parameters k„. CURVATURE. the period equation is derived from the condition that the determinant formed by the boundary is conditions vanishes. and k that the velocity of propagation around a cylindrical surface depends plotted in Fig. a good approximation may be obtained by considering an incompressible medium. It is seen that the curvature produces an increase of phase velocity mth wavelength. AND VISCOSITY 265 In contrast to the nondispersive Rayleigh waves associated with a plane surface. and "slow" waves controlled primarily by gravity. Since Rayleigh waves are affected only sHghtly by compressibihty. Eq.0 0. kp. 1.28. and experimental results are in agreement. As in the preceding chapters. the difficulty of the problem increased greatly for a sphere in that the solutions take the form of spherical harmonics. Love primarily [29] investigated this problem in detail and found that when applied to the earth the solution indicated "quick" waves. 5-1. first derived by Bromwich [3] and later obtained by Love [29]: .0 'Ika 1. For a wavelength equal to half the radius of curvature the increase is about 10 per cent. To show the magnitude of this effect we have computed from Eq. given without numerical calculations. We give only the final result.25. and his experimental results are plotted in Fig. However. {Calculated by Oliver.5 27r. 5-1 a corrected Sezawa's curve upon the wavelength.28. for the case X The effect of cylindrical curvature on Rayleigh-wave propagation was [41]. Corrected phase-velocity curve of Rayleigh waves propagating around circum- ference of a homogeneous cylinder with Poisson's constant of 0. Rayleigh waves from an asymmetrical source were also investigated by Sezawa and Nishimura [55].(M J„+j being the Bessel function of the order (5-60) n + ^. e the azimuth. and we can put n = ka. Wa the radial. mA = ^1^ \f/n{kfia) kff (5^1) only and obtain Eq. U2.. Sato [48] considered the boundary conditions at the surface of the sphere in their most general form. 8 the colatitude. n is very large. Recently Jobert [21] reconsidered the Love-wave problem for a layered sphere and found that the velocity of the longer waves is significantly increased as a result of curvature. that the velocity of propagation of Love waves on a spherical surface is approximately equal to that on a plane surface. and by co. neglecting the effect of gravity. and azimuthal components of displacement. and take the polar axis to . ^ ^ where a is the radius of the sphere and Uha) = (-1)"(|)\m"^"'*V„. Sezawa found. For this case Bromwich derived an approximate formula (5-59) From Eq. For wavelengths small compared with the radius of curvature. may readily be obtained as in the case waves in the earth when the effect of curvawas further studied by several investigators. Sezawa [51.. Denote by R the radius. The index n is arbitrary and determines the mode of vibration. colatitudinal. In the second of these papers the propagation of Love waves in a surface layer overlying a spherical core was considered for the case of an asymmetrical source. coa. among other conclusions. effect of spherical which (5-25). 5-4. 56] formulated the problem of transmission of Rayleigh and Love waves on a spherical surface.:(fcga) ^ yfcâ (2n '^ rp„{kpa) n{n + + 1) 2) + ngpa/(2n + + ngpa/(2n + l)n 1)^ - feâ72(n kla/2{n - 1) 1) ^ . C03 the components of rotation. Ui.266 ELASTIC WAVES IN LAYERED MEDIA ^. Matumoto and Sato [32] recently discussed the transverse vibrations of a layered earth for two extreme cases. may be used to consider the of effect of gravity With the use model seismology techniques the curvature for compressible media of cylindrical curvature. solid mantle and rigid or hquid of seismic The propagation is ture also included core. one can obtain the dispersion of Rayleigh waves introduced by spherical curvature and gravity. the ratio 2Tra/n represents the wavelength 2Tr/k. Since only sectorial harmonics are involved in this solution. General Solutions for a Spherical Body. (1-13). (5-69) becomes 6 = R^s'm 8 d{uiR^ sm.Ur) dq. ae a/2 R d8 The equations may is Eqs. and we = div s = d{S2S.u. ^2 Thus Eq. q2. CURVATURE. R^ d8^ R~ Furthermore.u. In made of two formulas the line element is order to obtain Eqs.S^ dql + Si dql (5-67) and q. diUiRski 8) . diusR)] .- are unit vectors along the axes of curvilinear coordinates. obtain *Si 5(1^2^2) d(S.i R ' sin de 2/i d(Ru:i) Rs'm d^h ^ 5^' 8 de R aco. Qs.\_ + For spherical coordinates. are equivalent to w) the displacement and 6 be easily obtained from the vector form of = div s. Sz 1 = R dqi sin 8. use coordinates now be omitted.U. S2 = R. dR (5-62) _ ~ (X + 2m) dd 5 _ 2m a(fe) /? 2m '^ /2sin (5-62) V._ ^ 0) . (5-62) for generalized orthogonal q^. If given in the form ds' = Si dq] + . d(SsS. If s(u. these equations P "I^ at = (^ + 2m) grad div s + = m(V's - grad div s) + pF (5-03) Now and where o> is curl curl s grad div s — V^s (5-64) (5-65) curl s = 2(0 the rotation. d{S. 8) .) [ ^ ' ^ d(S. 267 The equations 2/i of motion then take the form 8) ^ ^" ^fc^sjn = ^^ + 2n b(ji2 ^"^ dR R sin h 2/x d8 â. Eq.THE EFFECTS OF GRAVITY. the curl of s can be written in the form curl 8 = d(S:iU:i) d(S2U2) 77^ S2S3 L dq dqs J + q2 djS.S2ih) (5-69) dqs S1S2S3 dq For spherical coordinates. AND VISCOSITY pass through the epicenter.) S1S2 L (5-68) dqi dq2 sin" 5 ds^ = dR'^ + = 1. dqi _ SsS.U2) + de. (X (5-63) may be written in the form P ^7^2 = at + 2m) grad ^ - 2m curl w + pF (5-66) The external force pF will is from (5-66). (5-62) by using Eqs. is formed of concentric layers of varying elastic constants X and He replaced the three equations of motion.R sin 2co2 R sin 5 a(M2-R) dR (5-71) 2ws ai2 as" We could eliminate and obtain four variables d. & are functions of A R as given by the equations 1) "^ JL r ^(^ + A . The A.. Jeans [17] assumed that the earth tx. and 2wi = 1 ^(wsi? sin b) dÛzR) de 0) R' sin 5 1 38 d(th.p{ six expression Q sin me (5-73) = P'::(cos 5) an associated Legendre fun(3tion. in the Ml Omitting the time factor exp form (io^t). • • • .^ ^' rr B = BG P C d(RG) ^ _ ^ ~ ^R'^ R dR and = dR^^ ~P BV ^ (5-74) ^ F{k^R) F(kpR) C'djRG) _ . the three equations for the variables solutions of Eqs. u^.268 ELASTIC WAVES IN LAYERED MEDIA. C03. (5-70) and (5-71) partial differential equations of the second order for the cog. (5-62) in a direct by and C03. w. as was shown by Sato [48]. U2. we can write the solutions of (5-62) = APZ = _ cos me + A'PZ sin ' m€ cos U2 d8 sm me 5 + U3 sm cos sin 5 me (5-72) = sm 5 do me _ sm where is P". We can also find the way. 6.^dR'^^ R ^ r ^ ^ . -Wi. 5 + 6'f.rF ~ ^ R'^ R dR (5-75) F = G = = ikMy'Hl%(kM) = ikeR)-"Hll\(hR) The Hankel function i/„+l is taken in order to have an outgoing wave. u^ in Eqs. in which the gravity terms were retained.. . (5-76). In a compressible planet which is in a state of equilibrium the "initial" stress can be assumed as an initial pressure determined by the condition of hydrostatic equilibrium. (1-7) written for the equilibrium of a perfect fluid (see Sec. Since it is assumed that a planet is formed by concentric layers. e) X ^"(cos X ^/"(cos 8) cos 5) cos me X X t-^^ (cos 8) sin f^^r'(cos 5) sin Then on writing solutions represented by (5-72) for for example. 1. €) W3 = U:. M = X) S ^ "^" cos me + X S -"^r^ first n sin me (5-78) we can compare (5-78) with the expression in Eqs. We shall assume that an undisturbed planet is a homogeneous body having a free spherical surface but for the sake of generality the density Po will at first be taken as a function of the distance R of a point from the center. Po grad To = grad po the (5-79) Now if be equal to the we assume that a disturbance occurs. is of equations will result of by means known at the surface. sum Vo + Vi. C. Now we 11^(8. AXD VISCOSITY These A. wave propagation in the interior of a gravitating compressible planet. instead of the displacement components. according to Eqs. as well as the case where the stress. Ref. set which the expressions AZ. The arbitrary constants are then found from (5-74). Then. CURVATURE. solutions • 269 {R = in terms of six arbitrary constants can assume that on the surface of the sphere a) the displacements have given values are expressed • • . its gravitational potential Vo will be a function of R. Wave Propagation in a together than to consider Gravitating Compressible Planet. all subscripts m and n and taking the sums. 1-2). CIT may be determined. This was done by Love [29]. Wi t/i. B. e) = = UsiS. and f/3 are arbitrary functions and can be expanded into Fourier series. It is more natural to consider the effects of gravity and curvature of the earth who investigated the laws of them separately. • • A . 22) can be discussed in a similar way.{q. = e) U2 = U2U. where Vi is due new potential T^ will to the disturbing forces as . e) = XI t/"(cos 5) cos me me + + + XI U'^'{cob 8) sin me me me (5-77) 1/2(8. as follows: C/i(5. e) (5-70) U2.THE EFFECTS OF GRAVITY. It was pointed out by Sato that the initial-value-problem studies by Homma (Chap. if given by Love first we neglect the products of small quantities of the order u^ (5-82) 6.270 ELASTIC WAVES IN LAYERED MEDIA well as to the change in the distribution of density. (X + M)^ + \ < ^^ v-72 MVt.. z) in the strained state. the density p of a particle when P it is displaced will be P= if po is -'^= Po. and Eqs. = -(po - s ^) + Xe + 2m I' P. and p„ at M{x. y. • • • (5-84) There are no changes take the form in the shear components p^^. p^y. (1-7) Po^ d u = . To simplify the theory. Let p be the density at a point M{x. By and Poisson's equation V^V = —4:irfp. and po have everywhere the same value.„ = • • • P„ = p„. Then. it will be assumed that X.PoO-sjj^ (5-81) taken at Moix — u. z) an instant t are equal to the sum of the equilibrium stresses and additional stresses in the disturbed state. Px„. and their derivatives. at P. and s the radial displacement. y. Since the cubical dilatation d = — (p' — same place. V^. Then s = w| + y| + w| pt^/po at the (5-80) Let p' be the density in the deformed state but at the initial position. z — w). n. y — v. where / is the constant . Finally. by (1-11). By (5-81) we have p = po(l - e) (5-82) The equation of the surface of a planet in the disturbed state can be written in the form R = where Ro So Ro + So (5-83) = = radius of undisturbed surface superficial displacement we assume that the stress components p^^. -Tx\F^-'m)^'^^^-^\^-^''^ ^•0 ^^-^^^ _ ^^ _ [29]. Ro In order to obtain an equation for the dilatation.THE EFFECTS OF GRAVITY.W). x. we obtain by using (5-86) 6£Po^ (5-91) Po = y df (X + 2^)V'9 - ^+ ^ V\Rs) + î? dR Ro Ro turn. CURVATURE. y. we have to eliminate the second term of the right side. the dilatation 6 vanishes. z in and applying the we obtain an equation which. On (5-87) with respect to the coordinates by differentiating Eqs. diyd) \ dz j (5-90) Thus the propagation if of rotation will be determined by a wave equation. (5-87) and taking into account that curl grad <p = 0. (5-85) can be transformed as follows: we is Vi d U Po (X + m) ^ + mVû dx dx (5-87) d2V Po dr Po Now and the g dVo dR = o TrfpoRo (5-88) effect of gravity is represented by two terms in Eqs. On operating with curl on Eqs. the gravity and pressure terms in Eqs.{Rl . (5-87). AND VISCOSITY of gravitation. The potential of a homogeneous sphere is F„ = 27rfp. leads to jLtV — Pq ar V^ - Po ^+ "o" ^/PO d- + ^ttK i^lV 2_d dR' R dR 6 = (5-92) . we obtain three equations for the components of rotation: dw 0. and adding. adding the results. Multiplying both sides of the three equations (5-87) by operation V". (X 42jLi) together with (5-91). +3 7r/po|-the other hand. n„ = 1 du dz dw dx O ' dy write the first of — ~ i 2 Ua: (— _ ~ — dy (5-89) We these equations in the form Po dr MVf2. By (5-79) and R' = x' + 1/ + z'. = If 47r/po0 (5-86) consider wave propagation and neglect external forces. 271 we obtain V'7. the term due to changes in density only. the attenuation is slight.^ (dw . and a are constants. ously. dll\ These expressions. (5-90) and (5-92) From is we surmise that when the effect of gravity- included the separate existence of distortional and compressional waves does not occur. Voigt Solid. k.^^^^—^-} It is seen that the effect of gravity is to introduce a very slight dispersion depending on 5-5. the effect of internal friction is to produce attenuation and dispersion of elastic waves. For a discussion of internal friction the reader is referred to the work of Kolsky [25]. because of the conversion of elastic energy to heat. When constants corresponding to the earth are inserted in these equations. relations of Sec.. Several mechanisms have been proposed for energy dissipation in vibrating solids. Love found that. Effect of Internal Friction. In practice. the stress-strain = X0 + 2m -dz + . dzdt at ^^_g. 1-2 take the According to Voigt's definition form p. in (5-92). In all problems considered previmedia were assumed. du\ . Substituting 6 = A cos k{x — at). however. where A.){Rlk'y^ are neglected. The effect of internal friction is more pronounced for higher-frequency explosiongenerated elastic waves where it may influence the shape of the elastic pulse. the velocity of compressional is waves a = ao-\- 8(x ao = V(X + 2m) /po (5-93) where the correction factor 8a 2poao5a is obtained from (5-94) = -^^ ^4 .272 ELASTIC WAVES IN LAYERED MEDIA Eqs. There sionally is ally convenient fit no satisfactory theory of internal friction. Several mathematicmechanisms have been suggested. and these may be grouped collectively under internal friction. Love showed that propagation is essentially the same as would occur if gravity and initial stress were neglected. however. and the dispersion is negligible for earthquake waves. d (dw . [64]. perfectly elastic The In general. may pi be obij^' tained by using in (1-11) the operator X ^- X' d/dt for X and -}- d/dt . (1-11) for M- for an elastic soHd. however. when 2ir/k is small and quantities of the second order in {gpQRo/iJ. similar to Eqs. It is well known. locality. that dissipation accompanies vibrations in solid media. which occaexperimental data over a limited range of frequencies. X' — + 2m' ^~r. Eq. (5-101) and (5-98) find (admitting only the negative root wave.'j. 273 of (1-13). This equation can be satisfied for complex f Inserting ^ Eq. (5-97) leads to the expression (5-99) P P . increases with frequency. leaving only a single constant A. we obtain the equations motion in the = W+ m) + (X' + m') 4^K^ dx dt\ (m + m' -^. CURVATURE. as given by the factor exp ( — rx). (5-99) and equating real and imaginary terms. the four constants solid. for r) that the attenuation of the . ii.Wu dt (5-96) Sezawa [52] gave these equations in a somewhat the wave equations take the form P different form. AND VISCOSITY Using these operators in form P ^-^. Similarly. the attenuation then being complete.1 THE EFFECTS OF GRAVITY. for a plane shear wave again propagating Substitution in the second of Eqs. \x must be used to specify a Voigt but simplifying assumptions can also be made. The phase velocity c = u/k obtained from (5-100) has the value (m/p) ^ appropriate for an elastic body when co = 0. phase velocity increases. = X + f/i will vanish by taking X' = In general. we can use the solution i/' = Ae" (w(-fi) (5-98) in the positive x direction. X'. To consider the effect of viscoelasticity of the Voigt type on a plane wave. becoming infinite with CO. + ii for the effect of viscosity. the "dilatational viscosity" (X' |m') which corresponds to the bulk modulus — f û'. For example. For increasing frequency. we find k' = k -\- ir in = 2(i/ HP</ j 1 + + m'V) m'V) we + ^-'f M _ + 1 (5-100) \ L Mpco^ / \L (5-101) 2(m^ M J From Eqs. = ll at ^^^ (X + / 2m) W+ 'V72 (X' + 2/) V^ ^ at (5-97) V72 I î X. in/lN (5-104) /r where s? = ^ — ^2 2 ' . . the subscripts 1 and 2 refer to a layer and an infinite substratum.— 274 ELASTIC WAVES IN LAYERED MEDIA period equation for Love waves in a Voigt solid The If may be derived following the procedure of Sec.^ 2 _ M2 P20^' + ^M2 Si and S2 being coefficients of z in expressions for iî and V2. the wave equations are d Vi (5-102) (m2 + M. respectively. The propagation of plane Rayleigh waves in a Voigt-solid half space was also discussed by Caloi [6].' |V V2 f} = P2 -TT2 Using the boundary conditions in the form ("' + "^^1 Vi = ^'^ = « (5-103) = V2 Sezawa and Kanai [58] derived a generalized Love-wave period equation tan . with trigonometric functions for the layer and exponential functions for the substratum. respectively. 4-5. who gave a generalization of Rayleigh's equation (2-28). „ Si/i „ TT = (m2 7 (Mi + — TT+ Za)Mi)Si 1 0^(^2)82 . When the real and imaginary parts in Eqs. ••• (5-107) poj ~ — X + 1 2m 2 + MX' + / (5-108) 2m') and POJ • (5-109) . two equations are obtained to determine a complex k. Eqs. (1-24) take the form (V' iv' where + kl)d = + kl)n^ = 2 (5-106) 0. / — A. where co can be complex in the general case. (5-105) are separated. For a time factor of the form exp (icot). 0 in sec. Equations (5-108) and (5-109) are obviously a generalization of the factors ka = oi/oc and k^ = co/^S introduced in Lamb's problem (Chap. cr = 1/4. CURVATURE. Phase velocity of Rayleigh (2) viscoelastic half waves in (1) elastic half space with m' = 0. this equation for a Voigt-solid We can. <n T Fig. k^/kl = 1 . therefore. = o (5-111) The real and imaginary parts of the roots of Rayleigh's equation were calculated for the values of m/m' = 30.5 1. (3) m/m' = 50.0 1—1 d 0. 100 sec~\ Caloi's curves show the dispersion and absorption of Rayleigh waves due to viscoelasticity of the Voigt type (Figs.5 |*h- 1. The wave number k now also becomes complex but the formal derivation of Rayleigh's equation follows the usual pattern. 50. ka where /c„ and k^ arfe now numerical calculations Caloi given by Eqs. obtain half space from Eq. 2). 5-2 2. 5-2. (5-108) and (5-109). (After Cahi. For his [6] solved this equation assuming that X = fx and X' = -Im'.) . The substitution kl/k^ reduces (5-110) to the form = 5 + 8/3.THE EFFECTS OF GRAVITY. AND VISCOSITY 27. using the transformation CO ' kff = k 16 We obtain 1 - + 24-16 ka_ k~B - kl 1 = kl (5-110. (2-29j. space with m/m' = 100.5 0.0 1. (4) fi/fi' = 30.^/16 B' + --f>+(i--f) and 5-3). both dispersion and absorption occur for this case. Using the methods of Chaps. 2-3 for a Voigt-type solid in which X' and m' vary as |co|". and Rayleigh waves. Absorption coefficient (3 Kq of in a viscoelastic half space = 3. 2. [35].276 19.99 ELASTIC WAVES IN LAYERED MEDIA 9. As would be expected. Newlands [37] generalized According to Maxwell's definition strain relations take forms such as Maxwell Solid.3 km/sec and n/ix' = 50. she obtained the characteristics of P.) Lamb's problem of Sec. S.99 ~ I 0.99 " i i< ) 0.09 10 20 Seconds Rayleigh waves 30 40 with Fig. 3. 5-3. {After Caloi. and 4. the stress- Pry di ~ ^^ dt (5-112) 8 . replace /z by /x/(l + l/iwd).For propagation of a disturbance in a Maxwell solid. co//c —^ -V'ZfjLud/p as co5 —^ 0. with the use of we note n in (1-11). AND VISCOSITY where d is 277 such a substance the the relaxation tlmf of the solid. CURVATURE.j THE EFFECTS OF GRAVITY. will flow indefinitely. approximating a viscous liquid. Internal Friction in Earth Materials. where wVf^ = m/p. From the form of (5-1 12 that. yi'oi/ix with quantities (= 1/Q) mentioned earlier in this section being surprisingly independent of frequency. the deformation being irrecoverable. For wave propagation it is related to the absorption coefficient by r = irf/Qc. Birch [2] reports values of l/Q = 170 X 10"^ . Attenuation is difficult to evaluate because allowance mission across interfaces. pressure. consider the propagation of a plane shear wave given by exp [i{wt — ^x)]. (5-114) we conclude that the effect of viscoelasticity of the Maxwell type on plane shear waves is to introduce an attenuation given by exp ( — Tx). which gives the equation This equation is satisfied for complex f = /c + it if i + 1 .^ 4.5 CO - 1 From Eqs. where r increases as w' for co 3 1 and approaches zero as 0)8 becomes larg e. Ricker [43-45] discussed this problem in a series of papers for internal friction of the type described by Stokes' differential equation. the behavior of a an operator such as m/[1 + l/d(d/dl)] for Maxwell solid can be found approximately from expressions derived under the assumption of perfect elasticity. For short-period stress variations Maxwell solid behaves as if perfectly elastic. must be made for energy loss upon transRicker [45] used a homogeneous section of shale to study the alteration of a seismic pulse. The velocity co//c -^ \î^/p as coS —> 0° On the other « . and temperature. Transient seismic pulses generated by earthquakes or explosions will be altered by this effect. In the foregoing sections it was on steady-state plane waves was to introduce frequency selective absorption and dispersion./i + 4^ (5-114) 2 T = po) 4- 2. For lorif^-continued stresses. There such as is increasing evidence that for a wide range of frequencies internal friction in crystalline rock is principally of the Voigt type. shown that the effect of internal friction Dispersion introduced by internal friction of consolidated rock seems to be negligible for frequencies under 100 cycles/sec. As an example. The dissipation function 1/Q is related to the logarithmic decrement A of free vibrations by the relation A = ir/Q. hand. Ewing. London. London Phys. 4. central 7. M. 6. Soc. pp. Comportement des ondes de Rayleigh dans un sHsm. lar. I' A. 52. Bull. L. Bromwich. 1906. pp. Mag. Geol. BuUen. M. pp. vol. F. Ice: II. pp. 1952.: Internal Friction of Solids. 1923. Sykes. 24. pp. 1937. Acoust. P. 98-120. vol. 204. Amer. Since the corresponding wavelengths are large compared with any of the discontinuities encountered by these waves. 10.. 1934. 1948. Spec. A. P.R. V.: 2. pp. Cambridge 5.: Love Waves in a Surface Layer of Varying Thickness. Some Recent Devel. Crary Propagation of Elastic Waves in pp. et 1953. A... the effects of scattering and refraction are minimized. 408-410. P. 1950. 276-283. and A. 13. Barnes: The Reflection of a Pulse by a Spherical Surface." 2d University Press. Roy. E. Jeans. X 10"^ at 4. 701-713. Proc. Soc. Mechanics Revs. 8.: Longitudinal Wave Transmission and Impact. Soc. and C. It is surprising that values of l/Q determined in this way for the upper mantle are of the same order of magnitude as those found in audio-frequency vibration measurements on crystalline rock despite the difference in physical conditions. vol. pp.: A Method of Measuring Some Dynamic Elastic Constants and Its Application to the Study of High Polymers. 9. pp. London Math. REFERENCES 1. Soc. /. Martin: Wave Effects in Isolation Mounts. R.. vol. 1898. 23. J. London. 17. 62. 1-3. D. Amer. Proc. milieu firmo-elastique Paris. al. Appl. 6. (London). vol.000 Ewing and Press [12] used the attenuation of mantle Rayleigh waves having periods of several hundred seconds to deduce the value of l/Q for the upper mantle.278 at 1 ELASTIC WAVES IN LAYERED MEDIA atm for diabase...: Recherches sur I'elasticite. in Particuon the Vibrations of an Elastic Globe. pp. 24. Amer. Seism. 44.: The Propagation of Earthquake Waves. 16. vol. 1954. icole normale 11. 89-108. vol. Trans. H. Harrison. 9-14. K.. /. milieux visqueux et non-visqueux. A. 1949. O. W. pp. 1952. L. 24. ed. Kolsky: An Investigation of the Dynamic Elastic Properties of Some High Polymers. 111-121.: "An Introduction to the Theory of Seismology. 5. H. 12. RheoL. Geophys. and H. vol. Donnell. C. Pubis. M. These are audio-frequency determinations on laboratory samples using longitudinal and atm and 280 torsional-free vibrations. M.. vol. United Trade Press. intern. R. 169-222. Caloi. Proc. S. London Phys. vol. 1953. Northwood. 1952. pp. 3. Soc. Davies.. bur. Amer. 62. Handbook : of Physical Constants. vol. vol.: indefini. 554-574. 14. B. 102. Press: Mantle Rayleigh Waves from the Kamchatka Earthquake of November 4. Homma. Davies. Soc. B. 181-184. pp. D. Acoust. On the Influence of Gravity on Elastic Waves and. K. K. 17. pp. M. Ann. 1949. 153-167. J. 27-37. .: Stress Waves in Solids. ASME. Soc. 1953. Proc. Hillier. {Tokyo). T. and F. 15. 30. Duhem.. respectively.: : Sur la propagation du mouvement dans les milieux visqueux. Hillier. Birch. Anderson. vol. 1952. pp. Cagniard. 62-71. Soc.. Paper 36. W. T. Ewing. Physics. 471-479. 1930. : IV: Proprietes generales des ondes dans les [3]. and M. 32. 31-30. vol. 28. Bull.: Notices Roy.: "Stress 26. Bull. Acoust. 1944. O'Neil. 1944. vol. 21. Dover Publications. 270-292. On Attenuation of Waves Produced in Visco-elastic Materials. Bull. 1943. Suppl. W. L. 1942. 26. Soc Amer.: Sull'impossibilit^ di propagazioni reale accad. and Duwez: The Propagation of Plastic Deformation in Solids.: Field and Impedance of an Oscillating Sphere in a Viscoelastic Medium with an Application to Biophysics. la 36. H. Ricker. 1952. 114.: Water Waves on Sloping Beaches. Amer. NDRC : Rept. Rayleigh Waves on a Cylindrical Curved Surface. 155-160. H. 1955 (in French). A. . Rend." Cambridge University Press.THE EFFECTS OF GRAVITY. Union.. 688-691. Matuzawa. H. 11. 39. Soc: Geophys. 27. 25. Acoust. vol. The Effect of Solid Viscosity of Surface Layer on the Earthquake Movements. vol. Lewy. pp. 1947...: S-Wellen an der fest-fliissigen Schichtgrenze: I. Jobert. E. J. 369-406. Lincei vol." 4th ed.. 2. 1890. 2. 1931.: H. Soc. Von Kdrmdn. pp. pp. 19.. Phys. 1953. Mattice. T. Jeffreys. 2. M. Phys. Further Developments in the Wavelet Theory of Seismogram Structure. pp. I. Japan. Lamb. Earthquake 31. vol.318-32:^.: Reflection and Refraction : Bull. pp. /. vol. 987-994. Earthquake Research Inst. Oestreicher. 1954. geophys. 40. (Tokyo). 23. 24. 1931. of Plane Shear Waves in Viscoelastic Media. Movement vol. Oxford. ponts et chaussies. 27-73. 194-201." 6th ed. C. Earthquake Research Inst. 1954. 1950. New York. Dover Publications. and K. 42. T. pp. naz. and Y. 434-448. Ajiier. On the Propagation of Plastic Deformation in Solids. J. Jeffreys. Waves in Solids. A. 23. 26. 52. Bull. London. Kanai.: "Some Problems of Geodynamics. E. 1954. pp. pp. Von Kdrmdn. 1955. 365. H. T. pp. 33.. Y.: "The Earth. vol. Lieber: II. fluidi viscosi. pp. 1926. Rev. pp. and New P. Seism. Kolsky. vol. 1952. pp. 613-624. Soc." Cambridge University Press.. vol.: Mouvements ondulatoires de croissante. 34.: "Hydrodynamics. Maxwell.. 247-258. 1950. 24-25. /. Astron.. 35. 19. pp. 33. 13. H. 32." Clarendon Press. 279 H. ondose nei [6]. mer en profondeur constante ou de- Ann. 29.. 1931. OSRD T. 1954. in Seismograms. Earthquake Research Inst. Lampariello. Ayner. Soc: Geophys. 1946. Matuzawa. pp. 1-6.. vol. 1945. Anlron. Trans. 928-935.. Soc. 204-211. Cambridge University Press. York. 32. K.: iS-Wellen an der fest-fliissigen Schichtgrenze: p. CURVATURE. vol. C: "Scientific Papers. Miche. T. (Tokyo). 35. Soc. Nomura. J.: Acoustic Determination Materials. Bull. 32. 28. 43. 1952. A-29. A. pp. On the Cause of Oscillatory . 20. Sato: On the Vibration of an Elastic Globe with One Layer: The Vibration of the First Class. . 1-48. pp. 197-228. vol. (Tokyo). Geophys. AND VISCOSITY 18. 1949. H. 75. P. Suppl. Acoust. 21. 22. vol. Oliver. 1911. 26.. Matumoto.: Lamb's Problem with Internal Dissipation: vol. Phys.: Damping in Bodily Seismic Waves. (Tokyo)." 3d ed. Am. Soc. Math. Ann. T. H. Jeffreys. A. Earthquake Notes. vol. 1954. of the Physical Constants of Rubberlike 38. 737-775. H. J. Newlands. Research Inst. Takaku: On the Propagation of Elastic Waves in an Isotropic Homogeneous Sphere. N. pp. vol. G. vol. 41. London. Am. 30. H. 7. Nolle. pp. Monthly 407-410. 131-164. London. Appl. Love.: vol. Monthlij Notices Hoy.: "A Treatise on the Mathematical Theory of Elasticity. pp. N.: Effect of the Earth's Curvature on Love Waves. 37.. 707- 714. Love. 1944. and K. Scholte. Bull. R. N. Akad. vol. Civil Engrs. 1892.. 50. Sezawa. : pp. vol. Acad. 47-62. 514-534. 2. Earthquake Research Inst. Sato. M^m. 437-456. 63. vol. vol.: 46. pp. Ned. 57. Paris. 1952. K. K. Stoker. Earthquake Research Inst.: Contribution a la theorie des ondes liquides de gravite en profondeur variable. vol. 43-54. Gewone Vergader. (Tokyo).: Further Studies on Rayleigh Waves Having Some Azimuthal Distribution. vol.R. 3. J. 60. and K. Sezawa. et. sci. Kanai: On Shallow Water Waves Transmitted in the Direction Parallel to a Sea Coast. 1929. Afdeel. 45. 1914. 314- 323. 55. 669-676. 486-519. M. Earthquake Research Inst. I. vol. Bull. 56. /. vol. Physik. Earthquake Research Inst. . Bull.: Boundary Conditions in the Problem of Generation of Elastic Waves. pp. Earthquake Research Inst. N. : 1938. Phil. 4. 1946. Wetenschap. tech. vol.: ELASTIC WAVES IN LAYERED MEDIA Wavelet Functions and Their Polj^nomials. 62.. C. 1931. 1-10. pp. 49.. 1-18. Bull. pp. pp. Sezawa. 231. Quart. 47. K. G. On the Propagation of Rayleigh Waves on Plane and Spherical SurEarthquake Research Inst. L. le 47. Dispersion of Elastic Waves Propagated on the Surface of Stratified Bodies and on Curved Surfaces. 7. Taylor. 1950-1952. vol. 17. 48. 339-407. pp. pp. Sezawa. Sezawa. Roy. 685-694. Bull. vol. (Tokijo). W. 1933. Natuurk. 5. Sezawa. Bull. Sci. World Petroleum Congr. 671-693. Earthquake Research Inst. (London). Geophysics. 1929. pp. (Tokyo). Bull. 9.. Kanai: Damping of Periodic Visco-elastic Waves.: Proc. Research Inst. 1927. G. Acad. K. Paris. Ann. (Tokyo). 1928. (Tokyo). Sezawa. (Tokyo). The Hague. Ricker. On the Theory of Visco-elasticity: A Thermodjmamical Treatment of Visco-elasticity and Some Problems of the Vibrations of Visco-elastic Solids. J.: The Testing : of Materials at High Rates of Loading. 1-18. Sezawa. 58. 16. Bull. 9. K. 1927. British Official Rept. K. I. 6.. J. K. The Reflection of the Elastic Waves Generated from an Internal Point 51. 61. Sur : mouvement des milieux visqueux et les quasi-ondes.: Over het Verband tussen Zeegolven en Microseismen: I. (Tokyo). and G. pp. K. presentis par divers savants. 53. pp. pp. K. pp. 123-130. pp. Math. pp. Pubis. J. no.: On the Decay of Waves in Visco-elastic Solid Bodies. Propagation of Love Waves on a Spherical Surface and Allied Problems. 27. 1947. Sci. (Tokyo). Taylor. 1940. 491-503. Sezawa. 59. Y. Nishimura: Elastic Equilibrium of a Spherical Body under Surface Tractions of a Certain Zonal and Azimuthal Distribution. 35. 1939. 52.280 44. H. 6. Earthquake Research Inst.: faces.: Surface Waves on Water of Variable Depth. A. 1951. 26. K. (Tokyo). Voigt. Inst. Ricker. 70. Bidl. pp. G. The Form and Laws of Propagation of Seismic Wavelets. pp. (Tokyo). Soc. : of a Sphere. with Special Preference to Love Waves in Heterogeneous Media. I. (London). Sezawa. 1943. and several notes in C. (Tokyo). Trans.: Propagation of Earth Waves from an Explosion. vol. 3. 1-54. Srd Congr. 1949. 64.. 1927. Ueber : innere Reibung fester Korper. 115-143. Verslag. pp. Bull. Roy.: On the Transmission of Seismic Waves on the Bottom Surface of an Ocean. ministere air 275. Earthquake Research Inst. 21-28. with Increase in Focal Distance. vol. Roseau. 2. Thompson. vol. vol. Earthquake Research Inst. vol. Bull. vol. 1929. Earthquake 54. vol. C. Appl. vol. 52. Then we have —H assuming. As which we can on the period equations from main characteristics of wave propagation. wave propagation in elastic plates and cylinders is analogous to propagation in layered spaces. and Sato [73]. of preceding chapters will again appear. T~ ax (6-1) and yp are solutions of the wave equations VV which satisfy four of the plate.„ .„ . The plate can have either finite or infinite dimensions. Oscillations of an elastic plate. The simplest case is obviously that of a homogeneous plate bounded by two parallel planes. To derive the period equation in this problem it is sufficient to consider the propagation of plane waves.CHAPTER 6 PLATES AND CYLINDERS In many respects. solutions of the form <P = (A sinh vz -\- B cosh vz)e'^"'~''''' '"-'''' . and more recently by Prescott [63]. were investigated by Rayleigh [71].. Gogoladze [24]. Lamb [41]. Procedures for developing solutions can be similar. and several of the wave types encountered in the layered-space problems before. as usual.v (0-4) rP = (Csinh/0 +D 281 cosh u'z)e" . = A^ H VV = ^^ (6-2) boundary conditions at the upper and lower surfaces These conditions express the fact that the stresses vanish and z = (the thickness of the plate is denoted by at the faces z = 2H and the median plane hy z = 0). and we now restrict ourselves to the latter case. 6-1. the surfaces of which are free of stresses. The displacements are written in the form most of the discussion will center infer the ^=n~a dx The potentials (p dip d\p az ^ = d(p r" oz + . dip . and others. Plate in a Vacuum. = m/p and inserting expressions (6-4) obtain the four boundary conditions in the form (fM - 2nk^{A sinh vH . we write the columns first line of and subtract the second from the fourth.B cosh vH) . Then.D sinh v'H) = 2ikv{A cosh vH — B sinh vH) -ifx/ (/' + A. They are - 2txU'W^ + k^) sinh j^// cosh v'H + and (pco' 4mA.^ „.')(Csinh v'H . V = Vk' - kl v' = Vk' we kl k^=a h = -^ (6-5) p Noting that a^ into Eqs. add the third. B. on = = 2iik'^ — pc/ = m(2/c' — = k^) = ii{v' -\- k) rfi-?") 2nkv' oosh v'H d 2nkv' sinJi v'H we have —a A = 2a cosh vH sinh vH -(v'^ id a cosh cosh v'H vH vH {v'^ —ib -2ikv cosh vH + k^) 2ikv sinh + k^) sinh v'H —2ib -2{v'^ -4ikv sinh vH + A:^) sinh v'H (6-8) Obviously this equation can be (pco' split into two. (6-1) = (X + 2n)/p.') (A sinh vH -\- B cosh vH) 2ikv{A cosh 2ifj. (6-6) - 2mA.kv'{C cosh v'H + D sinh v'H) = + Z) vH -{- B sinh vH) + The minant to the putting a h (/' + k^){C sinh v'H cosh v'H) = the deter- period equation is obtained in a simpler form if corresponding to the coefficients A. 'j'v' sinh vH cosh /// = (6-10) .D cosh p'H) = . C.2itikv'{C cosh v'H .282 ELASTIC WAVES IN LAYERED MEDIA where. 'j'j'' cosh i/// sinh v'H = (6-9) - 2mA-')(/' + k^) cosh j/i/ sinh /i7 + 4mA. D. 0^ and (6-3). from the wave equations. Rayleigh waves in an For a = .4kW = (6-17) This equation is recognized as the characteristic equation elastic half space discussed in Sec. and the left side of (6-12) {2k' - A. the asymptotic limits for long and short waves are first considered. Thus we can consider a motion which is given by symmetric with respect to the plane z = D <p = B coshpze'"''-'^^ yp = Csinh /2 e''"'"*"^ (6-13) and the antisymmetric motion represented by functions ip = A sinhvze' "''-''' yf^ = D cosh Zee' ^"'"'"^ (6-14) is determined by the correspondby (6-12) for the symmetric and by (6-11) for the antisymmetric case. and. For very short waves and c < < a. ing period equation.4kV = (6-15) By (6-5) we obtain c' J. ^'\ 4(X + m) cl . 3/3' = a.|)' .. v'H are becomes unity. ~ ~ + hy (2 . In both cases the nature of the vibrations i.. we have c^ = 2 \/2 a/3 = 2 V2/S0. where a Cp is the phase velocity of long longitudin al o r plate waves. 2-2.e. vH. The discussion of these transcendental equations in the general form presents certain difficulties. therefore. Then. two separate systems. Symmetric Vibrations (Mi). the quantities kH. It is easy to see that the coefficients A and can be separated from B and C. those equations take the form c'/ff tanh vH tanh i''H ^ 4Vl . TMien = I. if the hyperbolic functions are replaced by their arguments. giving large. v'H may be taken as small when c = o}/k is finite. For waves long compared with the thickness 2H the products kH. ^_.eiff iy''' (0-11) and tanh tanh vH V (/' H + hy 4kW (2 - cV/S')' 4 Vl - c'/J Vl - (6-12) c7/3^ its Now the transformation of the determinant is A which preceded representation in the form of a product (6-6) into equivalent to the spHttinj^ of Eqs. vH. (6-5). (2-28) for j.PLATES AND CYLINDERS 283 By Ecjs. (6-12) takes the form (/2 4- uy .^/a Vl ^ _4kW. The second mode M12 is typical of all higher modes in that c> (3. For c > /3. the waves described by (6-12) are dispersive. 6-1 (see Chap. 284 it ELASTIC WAVES IN LAYERED MEDIA was found that Cu (6-12) that c -^ ^ as = 0. 6-1. it may be verified from Eq.. 4. (After Tolstoy and Usdin. 0. and f or fci/ = ^ ^ . and p was given by Gogoladze Taking the variable = 1/c = k/co. c > ^ because of the periodic nature of the functions tanh v'H. Phase and group velocities obtained from Eq. the In general. of kH we can find a maximum and a minimum [24]. An infinite number of higher modes Mja. presented for the first two modes • • • Mn M^ \ c \-^l^^ ""TN^îs -^ ^ 2 Ref.9194/3. we can write the frequency equation . First and second symmetric modes il/n and M12 of a free elastic plate with Poisson's constant of 1/4.) mode the phase velocity decreases monotonically with increasing values of kH from c = Cp at kH = to c = Cjj at kH = 00 The group velocity has the same asymptotic limits but exhibits a minimum value when kH 4. A general analysis for arbitrary t? X. /x. kH ^ CO. For /ci7 00 c -^ t/ -> /3. For intermediate values value of group velocity. To determine manner in which the long. c -^ 00 and ?7 -> 0. 203). The lowest mode exists for which Cr ^ c < Cp. In the first ^___^ U /3 /3 ' 4 6 8 kH Fig. (6-12) and U = c -{.and short-wavelength limits are connected exists for one must use the complete equations.k dc/dk are and M12 in Fig. J^ - 1?' J^ - t?' tan IhcoJ-2 - ??0 (6-20) The expression (6-19) decreases in the interval (0. Dispersion occurs for these waves. There are no such roots in {l/cn. For waves long compared with the thickness 2H and c < ^ < a Eq. Again 0.f) (6-18) Then.^â which is obtained for uH = 0. This (6-17). When ojH —^ c» a real root of the period equation approaches ^r = l/c^ > 1//3. To discuss an entire dispersion curve. . after some algebraic increases. of roots of (6-18) is ^ Thus the number transformations. group velocity occurs at kH = 3. 6-2 given by Tolstoy and Usdin.= j is assumed.6. l/a).^^ 1/a . C/ —» (flexural waves). Antisymmetric Vibrations (M2). For kH —^0. 00). The number of modes corresponding to the phase velocities for ^ < 1/13 increases with increasing frequency. C/ = /3. It is seen that . One can consider in the plane {a.1) (6-21) In deriving (6-21). For kH -^ 00 and c < (3 < a. for the interval < < < 1//3 all factors are real. I/cr) there is a single value of coH which satisfies the period equation. with phase velocity decreasing to zero with increasing wave length. For intermediate kH a maximum value . For c > (3 and kH —^ <» c -^ . - ^= = now -4^^Vi^ t? ^^ ^!^-^' t^^ (//. the third terms in the expansion of the hyperbolic is the period equation for long flexural waves.h) ^^"^ (^" w " ^' ^ ^^~^^^ and <r = . There is also a minimum value t? = 1/2/3 Vl . for the lowest and of for kH -^ mode M21. since i^r = 1/cr = t?max (at (jiH = 00). (6-11) reduces to Rajdeigh's equation functions must be retained. (6-11) reduces. ^ c —^ U -^ Cr. computations based on Eq. Gogoladze proved that for each ?? in (1//3. Eq. These appear for the first two modes in Fig. while (6-20) determined by the number of points of intersection or by the number of asymptotes of (6-19) and (6-20). < c < Cr. to W = I (mil .8.PLATES AND CYLINDERS (6-12) for symmetric vibrations in the form 28-0 (2^' - ^J tan {Ho:^J~. (6-11) must be made. and the propagation degenerates to Rayleigh waves associated with both free surfaces. t?) the curves determined by the equations '^ = ^' (2^' . Mz^. Impulsive Sources. {After Tolstoy and Usdin. Tolstoy and Usdin discussed this in some detail. are 0. For the latter case. 4. and of excitation of the sequence of arrivals at a distant point will be determined by the group-velocity curves. In general. 6-2. the condition U -^ corresponds The higher modes 71^22. As in the preceding chapters. The degree waves in any given mode depends on the spectrum of the source and the excitation functions . is the limiting case of *SF or P waves normally incident and c -^ 00 kH upon the boundaries. as kH -^ 00 ^ and c —» <» as kH -^ ^ . the period equations (6-11) and (6-12) may be interpreted in terms of multiple-reflected. some information can be obtained from the group-velocity curves of Figs. c—>f/-î3 Both maximum and minimum values of group velocity are associated with the higher modes at intermediate wave lengths. Although a full discussion for this case requires a treatment analogous to those given in Chap. a source will stimulate waves associated with the various modes. ^ to the condition of zero energy transmission in the horizontal direction. Interpretation in Terms of P and SV Waves. We mention a few special cases only. reinforcing SV waves for a > c > /3 and SV and P waves for c > a > l3. First and second antisymmetric modes and Mti of a free elastic plate with Poisson's constant of 1/4. The conditions kH -^ 00 c -^ j8 correspond to SV waves which are multiple-reflected near grazing incidence.) all characterized byc>/5.286 ELASTIC WAVES IN LAYERED MEDIA \ M22 ^^^ J^^jy<^ ^^ ^^ 4 6 ilf 21 <?=Cb ^ 2 8 Fig. 6-1 and 6-2. ^y^^''''^*^^-^^ 9A'J ^^f'Af\f\j\/\'^^ 9. In Fig. using the elastic constants and thickness appropriate for the experimental plate.e. 6-0 the 287 model seismogram from an impulsive spark aluminum plate is presented (Press and Oliver [66]). (6-12) and (4-94). as maj^ be seen in a plate will be included in located in a liquid (see Sec.v-^/Wx 8.9" '-—^'Âr/\J\f\y\.4" ---•^/y^/\^y*\y > — ^. is quite satisfactory. were excited. 6-4..4" MW\/V\/\A/"''^V^ ^^^—^sj\pj\j\j\y^^ 10. The excitation and the frequency response of the detectors were such that only the lower frequencies of the M-î mode.) in. 1/32 from detector D.9'' 11 A" 11. 6-3.- 100MS Fig. That the dispersion is proper may be seen from a source at the surface of a thin d= 8. i.9 -'MA^/lA/X/^^ 10.9" — MWVV\A/\/\---^''''''^ #AA/\/V\AyX/'^ . Spark comparison of observed phase and group velocities as determined from the seismograms with the theoretical curves computed from Eqs. Flexural waves excited in a plate of source S is at distance d 24S-T aluminum. (After Press and Oliver. the flexural waves. 6-3). The agreement in Fig.. The the discussion of SH-wave propagation is more general case when the plate . thick.PLATES AND CYLINDERS for the plate. Plate in a Liquid. and Fay [20]. Osborne and Hart were principally concerned with the so-called waveguide problem (homogeneous case). 6-2. cently discussed field of by Sauter Stenzel [80] investigated the acoustical a point source in a layer with an acoustically "soft" or "hard" boundary. 4000 \ I. and 4. . 3. where other references are given. {After Press and Oliver. We restrict ourselves to the former case. The investigation of flexural motion of plates led Uflyand [87] to a system of equations analogous to those used by Timoshenko [85] in the problem of vibrations of a rod. Fay and Fortier [18]. but the results could be generalized for an impulsive source. The vibrations of a thin elastic plate were re[74]. 2. treating only steady-state solutions. Mindlin [53] showed how to reduce these equations to three wave equations under certain conditions. whereas the other investigators studied the problem of reflection and transmission of waves incident on the plate (inhomogeneous case). Sezawa and Nishimura [77] considered a plate in an infinite solid medium. microsec 80 100 Fig.) Other Investigations. These results were applied to the problem of reflection of flexural waves at the edge of a plate by Kane [36]. 2000 K \ s^ ) A ^ o 20 40 60 T. thick. Osborne and Hart [59]. Observed and theoretical phase and group velocities in a plate of 24S-T aluminum 1/32 in. The analogous problem of transmission and reflection of electromagnetic waves in plates was investigated by Heins and Carlson [28] and Heins [29]. using the methods of Chaps. 6-4. Osborne and Hart generalized the solution of Lamb [41] for a plate in a vacuum. The propagation of elastic waves in a system composed of an infinite plate bounded by two parallel planes and immersed in an infinite liquid was investigated by Reissner [72].288 6000 > ELASTIC WAVES IN LAYERED MEDIA s. L^. Then the 2A Vo Pq. L2.2e e (6-23) where iô = vk^ — âo (6-24) The two equations corresponding components to continuity of the normal displacement w are — voe~"'"Ao + v(A cosh v{A cosh vH — B sinh vH) + vH -\- ik{C sinh v'H .D cosh v'H) = = (6-25) B sinh vH) . 6-5). Xi 2=0 z=-H f2 Fig. 6-5. (6-6) in the by Lj. OiQ. Po. The z axis is directed downward. we have form L2 L4 + Pocoê """Ao = = = - (6-28) Poco'e"'"Mo + L3 (6-29) . additional potentials are introduced to describe the participation of the liquid. L3. Plate in a liquid.ik{C&mh v'H + D cosh v'H) (/x + vê"'" A2 (6-26) The normal Vzz stresses for the liquid v-72 = \ AoV <Po = = 0) are ^0 gwô 2 — Poco <Po for 2 = -H H (6-27) Pzz = — Poco (P2 for Denoting the left-side members four other boundary conditions Li of Eqs. "o- ^0 z=// ^\ P\< <^î. /î. Xo displacements are difo dx Wa d(Po dip2 dz dx Wo = dip2 (6-22) dz and the appropriate solutions (Po for the liquid are (P2 = AqB e = A. below and above the plate respectively (scê will Fig. Ml. The liquid now be denoted by the subscripts 2 and 0.PLATES AND CYLINDERS 289 To provide for continuity of normal strossos and displacornonts and vanishing tangential stresses at the boundaries of the plate. :k â. 3.3 + ^ ca -i. tI5 .:^ -^ m ric' •^ ri.290 ELASTIC WAVES IN LAYERED MEDIA CO O tl: ^ .> a. I A m O . + tI5 tl3 133 O O + I . of the determinant of the system of Eqs.vlW + ^{k' P/3>o ^") sinh vH sinh v'H = (6-35) For the antisymmetric case [Eqs. (6-35) and (6-36) define relations between . Eqs. (0-20) (if On rearrangement (6-28). The last terms in (6-35) and (6-36) represent a modification due to the presence of the Uquid. ylo = ^2.^-c.4kW sinh vH cosh v'H . ifo = Aoe""^"-'"'-'^' ^ B coshvze''""-'^' (6-33) <p.Again these represent symmetric and antisymmetric vibrations of the plate. The period equation corresponding to the symmetric case [Eqs. (6-4) can be divided into two groups.kv' sinh v'H = (6-31 -2ikv cosh vH {v'^ + U') cosh v'H and V niv'"^ sinh k'^) vH cosh ik sinh v'H poc/ + vH 2inkv' cosh v'H (v'^ = (6-32) -2ikv sinh vH + ^ k^) sinh v'H Thus the terms in Eqs. in Eq. (6-32) and (6-33)] takes the form (v" + k~y cosh vH sinh v'H . and cp = Aê'"'^''"'-"'' ^ê-^°^"'''"'-'^' = A = D smYivze''"'-"'' (6-34) coshv'ze''-'-'^' = rp with Ao = — ^2.J PLATES AND CYLINDERS 291 {f)-2o).^-. (6-12) and (6-11)]. (6-30). the period equation takes the form for the plate) we assume X = /i shown Hence vo V cosh k'} vH sinh ik cosh v'H Pq(/ fx(v'^ + uH 2ifj. Because oi k = w/c. and (6-29).^kW vl){k' cosh vH sinh v'H vH cosh v'H + The sum ^ PP Vq (/c^ - - v") cosh = (6-36) ^ ^ of the first two terms in each equation is the expression derived for the case of a free plate [Eqs. = ^^g-o.x) ^ _ Csinh/ê''"'-'=^^ is a solution with ^0 <p. (6-31) and (6-34)] we have (v" + /c')' sinh vH cosh v'H . (6-16)].y[i+^ 2 The real part of this i poapCoH p0' I 1 V4(l . system of four equations This two. k. For wavelengths large compared with the plate thickness. As in the case of the plate in a vacuum.292 ELASTIC WAVES IN LAYERED MEDIA of the three variables c. the hyperbofic functions in (6-35) can be replaced by unity or linear terms. however. This attenuation is due to the leakage of energy from the plate to the liquid. (6-37) and (6-38).and high-frequency limits of the lowest symmetric mode can be found from Eqs. and we obtain the approximation more =2Mi-f-.^Va) ^ a' (6-37) equation is identical with the velocity of long longi- [Eq. We can again obtain expressions for phase velocity from (6-35) and (6-36) vahd for very large and very small wavelengths before proceeding to the and w. or kH large. containing four unknowns. in the lowest mode Cr < c < c^. When terms in (6-35) and (6-36) last the complex. Symmetric Vibrations. Considering steel plates in water. and becomes vq Re c > ao. The low.. split into can be equation each that show can be satisfied if we assume both c and k are complex. For short wavelengths. leaving the real part of the phase velocity unchanged. Complex c and k indicate attenuation of the waves. For example. Osborne and Hart computed values of phase velocity for this mode corresponding to intermediate wavelengths and showed that the effect of the water is to add an attenuation. or kH small. the attenuation increasing with the magnitude of the imaginary component. Other modes are introduced. The last term represents the effect of the hquid on the propagation of Rayleigh waves.4kW + ^ (e - vl){k' - v") = (6-38) The first two terms may be recognized as the expression whose root gives the velocity of Rayleigh waves in a half space with the elastic properties of the plate. The imaginary component represents the attenuating effect of the liquid which for low frequencies increases as the frequency and vanishes as w —» 0. of real roots for k and c provided that w difficult computations for intermediate wavelengths. This effect is small for a steel plate in water. a mode exists such that when kH is large and c < a^ both (6-35) and (6-36) reduce to 41 1 - 4^^ ^-m-[^-m-$ n PoC + ^(1 -^) =0 ' " '^ (6-39) . any two These equations can yield pairs is real and c < uq. we use c < /3 and obtain the approxi- tudinal waves of a plate in a vacuum mation (k' + p'r . by the presence of the water. Antisymmetric Vibrations. which gives the phase velocity for long fiexural waves in a plate vacuum. is According to Osborne and Hart. the attenuation the higher symmetric (6-36) reduces to great for mo. and the angle of incidence of waves were experimentally observed.st of modes (c > /3). 6-3. SH by allowing m2 to vanish. thickness of a plate. (4-212) Waves. Equation (6—41) 1 satisfied if 2kH. 76] confirmed these theoretical and experimental results. Press and Ewing [65] extended in their results to include the effect of the air. (6-21). (6-36) reduces to an expression for Rayleigh waves (6-38) or for Stoneley waves (6-39). For short wavelengths Eq. The interesting phenomena which occur in the inhomogeneous problem of supersonic-wave transmission through plates immersed in liquids gave rise to the investigations of Cremer. Schoch [75. Cremer [9] has shown that for thin plates total transmission occurs at an angle of incidence such that the trace velocity (phase velocity) is equal to that of fiexural waves. Floating Ice Sheet. = sin d and = where n = 0. Yov long wavelengths. giving the velocity of Stoneley waves (Sec. with the addition of an imaginary component (small for a steel plate in water). or kll small. With the substitution for (6-41) m cos 6 = n/o (6-42) . Maxima of transmission which depend on frequency.ss than ao. 4 . 1. Gotz. Thus we find tan 2A:i7^|2 -1=0 is • • • (6-41) where jSi/c 2H is the thickness of the plate. For intermediate wavelengths the first mode is again similar to that found for a plate in a vacuum. Gotz [25] generalized this result for any maximum transmission and simphfied Reissner's formulas. Eq. The period equation for this case can be obtained from Eq. gation in which the solution These waves represent a special case of Love-wave propais unchanged by the presence or absence of liquid media above and below the plate.-\/c^/0\ — = fc n-K.3 k^Hii \ ^'^ a 1 +^^j (6-40) / p„_ This equation may be verified by setting po = and comparing it with Eq. 3-3) at a liquid-solid interface.' PLATES AND CYLINDERS 293 A root exists for c close to but l(. Other Investigations. 27r sin d/U we can write . as might be expected. A discussion of the propagation of elastic waves in a floating ice sheet was given independently by Press and Ewing [64] and by Sato [73] in which an infinite plate underlain by infinitely deep water was used to depict a floating ice sheet. and Schoch. (6-41) or (6^2) and (6-43) given in Fig. 6-6. • • = • . Phase. 1. . 6-6. > — oo -^ c For each of the modes. The wave train is infinitely long.294 ELASTIC WAVES IN LAYERED MEDIA is which the condition for constructive interference between multiple- reflected SH waves with incident angle The in Eq. 6-6. n = 1. the frequency of the arrivals decreases.and group-velocity curves for SH waves in a plate. The mode can be deduced from group-velocity curves such as those in Fig.5 Fig.1 0. As time progresses. derivation of a similar condition for a liquid layer was given From (6-41) or (6-42) we may derive for the group velocity U= ^s'm 9 = — c (6-43) From Eqs. The first arrivals are high-frequency waves which travel with the velocity jSi. (4-83). L^Secc)nd rr ode \ S i 10 U Pi - ] 7' 72Hf 0. lengths given as 2H/1 by 2H/1 . î f corresponding arrivals given point to at a sequence of ^. U ^ a given • • • . .2. It is seen that c -^ we may compute oo dispersion curves (and t/ — > 0) at discrete wave- c c Pi p irst FTlode -. the slowest waves having the lowest of This case is almost cutoff frequencies given by / = n^^/AH. d and wavelength Zo in the direction of propagation. H cosh v{H) + Q8 sinh p. and p2 be the densities of the for respectively. = OX ^ dz potentials must be solutions of the wave equations (4-8). = ^ + 1^ dz ». the period equation can be written in the form P{2Q where 8 + 8 cosh v. a^ the velocities of compressional waves. = (A = ^1 (Csinh Kz +B +D E cosh Viz)e'^"'~'"'^ cosh /i0)e" "'-*"' (6-47) The arbitrary coefficients A. and the resultant determinant must vanish.H sinh v[H + k'^y sinh ViH cosh viH — 4vy. and the interface hy z — H. The median plane of the ice sheet is the plane 2 = 0. Taking the solutions (fi of the (4-8). the assumption Xi = îl or a = I can be made for ice but the observed values of Poisson's constant for lake ice are actually higher. are determined by the boundary conditions (6-45). the velocity motion "' — ^^ _ ~ *^ _ ^^ ~ ^^ (6-44) „. the free surface is then given by z = —H.k^ sinh v. of distortional Let p. waves in ice. and the boundary conditions are of the type (4-4). A homogeneous system of five equations is obtained in upon substituting the expressions (6-47) (6-45).H sinh ^[H = (6-48) = P20cl(i'[^ — k^)(vl — k"^) A PlPlJ'2 p = Q = (v(2 (v[^ + ky cosh v.e~ = Vk' .kl wave equations sinh PiZ v^ = Vk' we have kl.H cosh viH 4:Vivik^ (6-49) cosh v^H sinh v{H .. Under these conditions. • . We have ^^^ ^^ SV "' and P ice and water. B. we put vi = Vk' . To simplify the results.)i =0 = at2 (p„)2 = -H atz (6-45) = W2 (pzx)i (p„)i = H (6-46) By (4-10) I'. Thus we have the boundary conditions in the form The (pji Wi = = (p. and /3. SV and P Waves.PLATES AND CYLINDERS identical to that of electromagnetic 295 rectangular waves in wave guides (Terman [82]). 300 ft/sec. j8i = 6. (6-48).296 If these ELASTIC WAVES IN LAYERED MEDIA equations are compared with those given by Lamb [Eqs. Equation (6-50) corresponds to Rayleigh waves propagated along the free surface of a semi-infinite solid medium [Eq.8 cosh viH cosh v[H = 0. (6-38) derived by Osborne and Hart for a plate in a liquid and corresponds to Rayleigh waves propagated along the interface between a liquid and solid. (6-11) and (6-12)]. for a plate in a vacuum. a.9194/3i. unlike these cases.800 ft/sec. both of semi-infinite extent.[' + k'Y - 4v. which also represent the symmetric and antisymmetric solutions. In the first case we have P Q. By following Osborne and Hart. e < 1. some energy is radiated from the solid. or 5 is a modification term.e + 5 (6-51) The products of hyperbolic functions having large and approximately equal values were canceled. = reduces to ii. a^ = 4. The evaluation of phase velocity from (6-48) can be simplified in the limiting cases of very small wavelengths compared with the thickness of the ice sheet (kH = 2tH/1 very large) and of very large wavelengths compared with H (kH small). respectively. If 6 is small. Since their velocity exceeds that of sound in water. introduced by the presence which vanishes as the density of the liquid approaches zero.v{e = = (6-50) and {v[' + kY - 4vy. P2 1^ _ Cr Cr /3t —I Pi 2 1 2 Cr 2/3i/ Pi 4l5| 0-2 - 1 Cr 1 /3f i r 2 „2 Cr 1 2 i 2 Cr 2 "l c'r Cr /3? - (6-52) ai 1 2 1 - OClJ /3?J Using pi = 0. an approximation for the phase velocity can be deduced from (6-51) by substituting c = C/j (1 + e). it will be seen that P = and Q = represent the symmetric and antisymmetric solutions. respectively. It is evident that. The values c ^ ai and c = /3i are solutions of Eq. and (6-48) It can also be seen that of the liquid. a liquid.917p2. we may write the approximation equation is c« = 0. The root of this Equation (6-51) is identical with Eq. (6-35) and (6-36) for the case of a plate in P + 5 sinh viH sinh viH = and Q -\. = V^Pu we . the motion in a floating ice sheet cannot be reduced to purely symmetric and antisymmetric modes. Osborne and Hart obtained Eqs. (2-28)]. obtain = 0. + Then c (l - 4)* ^4 = we (6-53) solving for c is with the constants for ice listed above. Recalling from the previous section that the vanishing of the second factor in the period equation gives flexural waves.. In general. (6-54) for flexural waves is we find that by retaining terms to the order (kHY in this factor. the observed dispersion of flexural waves generated by explosions in floating ice is shown to be in reasonable agreement with the theory in these papers.87a2. In order to study the period equation (6-48) when in the kH is small. we may write Eq. 36) and extended by Press and Ewing [64]. (6-21) for a plate in a can be reduced to the corresponding expression for a plate in a liquid simply by replacing p2 by 2 p. The steady-state plane-wave theory of long flexural waves in an ice sheet on water of either finite or infinite depth was given bj' Ewing and Crarj(Chap. Solving for c^/(3l. = this reduces to the corresponding It is interesting to note that (6-56) Eq. the approximate form of Eq.y('-i). Ref. Their amplitudes decrease with distance from the write it interface.PLATES AND CYLINDERS obtain approximately e 297 = t'/4. the approxiin mation is much bettor with e = i/100. (6-40) c^//3^ power in kH. 3-3) traveling along the ice-water interface. For a steel half space and water. we form P{2Q + 8 cosh viH cosh p[H) = -Q8 sinh v.. we find /3i 3p2 1 _i_ 2 ^'^-'^^-' P2 For P2 vacuum. With this substitution we find that the lowest order of the right-hand member is (kHY^ and the lowest order of the lefthand member is (fci/)^". Using (6-46) and w = ck. 4. provided that we use no terms of higher power than {kHy*.H sinh p[H (6-54) Expanding the hyperbolic functions to terms of the third for flexural Avaves we use the fact that in Eq. . We therefore may take the left-hand member to approach zero. was found to be proportional to (kHy. (6-51 j the form (-i)i(-f-<-f.This the speed of Stoneley waves (see Sec. wdv e 0.8 \^ V.040 ft/sec. 6-7. V(5lO :it y2 Rif = 2. (6-54) is mode which analogous to the flexural in Fig.2 .and group-velocity curves from Eq. = 12. J Air coupir . and divide by v[kH. .and group-velocity curves for a floating ice sheet.298 ELASTIC WAVES IN LAYERED MEDIA [8] Crary for the computed phase. 1. - — \ 0. 0. These curves.5 1 10 2kU Fig. = 4.4 / / / / V 0. are based mode on the for a plate in a elastic constants vacuum. which is complex for . <r = 0.345 for ice. a. kH.1 0. (6-48) p 2Q Qb -\- sinh V]H sinh v[H b _ _ cosh viH cosh v\H of (6-57) expand the functions in powers is of the form 1 -f {kHY -|• • • . shown for ice and sea water. Crary.30.400 ft/sec. a. where form c^//3i is not a small quantity.5 1 f c y / ^ - / 0.9 Ma> . The first term and the second. 6-7.u JU 0.0 1 1 Flexural wave u.1 > / / / / / / 0.3 / 0.) /3 = 6.u . {Ajter in the For plate waves. Theoretical phase. we write Eq.7 / 1 / \ •^ 0.800 ft/sec for water.6 0. [8] An in seismic experiments unusual type of *SF wave was discovered by Crary on the floating ice island T-3.PLATES AND CYLINDERS c 299 > 0:2. which for large wavelengths increases as the inverse cube of wavelength. The solution for a plate in a these two expressions. the SV wave is totally reflected in its original form. no energy losses due to radiation into the water occur. recording is on longitudinal horizontal seismograph. we have replaced c^ by cl in the imaginary term. we from the plate to the liquid. These waves are propagated by multiple reflection of SV waves arri\dng at the ice boundaries with an angle of incidence ^cr = sin~^ /Si/ai. (6-48) is very difficult. indicating attenuation due to radiation from of long longitudinal of the ice sheet into water. 34). Crary Waves. 2. 6-8): 1. and the waves are propagated as a dispersive unattenuated train.ss of energy velocity. is of the order {kliy + iikK)' The terms containing {kHY introduce a slight dispersive correction to the velocity Cp. Crary. increasing very slightlj^ with time. Neglecting the find the approximation $\ 1 slight change in = 2 1 4- 2i{kHy 0C2P2 ^t 2\i 1 - 4/3? 1 - /5? (6-58) In deriving this equation. 4. vacuum is given by Eq. for c > a^. the tively attenuated. and Thorne (Chap. amplitudes are 4. For intermediate wave lengths the evaluation of phase velocity from Eq. Both real and imaginary components of phase velocity depend on frequency. 6-9. much smaller on vertical and practically absent on transverse horizontal. waves being dispersive as well as selecFor c < a2. (6-16). Phase velocity Travel time Principal is is near the speed of compressional waves in the plate. and the vertical displacement at the surface is zero. the phase velocity has an imaginary component. This is in agreement with the experimental work of Ewing. This wave has the following characteristics (Fig. The frequency is determined by the requirement for constructive .At this angle. as may be seen from Fig. 3. If we compare we can easily see that the presence of the liquid hardly affects the real part of the phase velocity but adds a small attenuation. intermediate between that for P waves and that for SH waves. Frequency is almost constant. (6-58). in which it was established that the velocity waves in floating lake ice was given by the real part Eq. and the term i{kHY represents attenuation due to lo. In general. Ref. 5 7 JUNE 1952 94 LB.7 OB 0.r - I ^ nl n = I.605 10.575 FT 10. showing P.1 I I i I AFTER EXPLOSION 02 Q3 0.4 I 05 I 0.200 10.7 0. The Crary waves are the large trains of nearly sinusoidal waves. TNT 02 03 04 05 06 0. (6-59) reduces to (6-60) 2H = On the ice island Crary found a. 2.1 NO. (6-59) arises from the reversal in phase on reflection interfaces. Eq.605 10. Crary. /3i / cos dcr = = 12.400 ft/sec 6.635 10.V 300 ELASTIC WAVES IN LAYERED MEDIA SECONDS 4*j046 0. NO.6 0. Seismogram obtained on floating ice island. Eq. 51 03:30 Z 31 MAY 1952 42LB. / being the frequency.TNT 0. and SH waves. SV.8 OS 'P r»Wii*i' SHOT Fig. /3i and the water-ice For n = 1.040 ft/sec / = 40 cycles/sec . (After Crary.0 —V -HT -HL — -HL -HT -HL V — AT 7575 FT.9 1.7575 7665 8200 7705 7705 7735 7735 SHOT 9 ' NO. (6-59) = (Sjf is The second term at the air-ice in the wavelength along the path. 7 1 20:30 Z 26 JUNE 1952 150 LB.1 02 03 04 05 I I I 0. 52 02:30 Z 1.665 1.8 " 09 '? +^ I -HT -HL V -HL -HT -HL --V — -V AT 10.635 — I SHOT 9 0.6 07 0.575 10.) interference AH where I cos e. 6-8. U 301 N L 1 / >v 1 ^P reflected \. of the plate Solving for the ice thickness 2H. or in water beneath a floating ice sheet.4 ^-^ 0.1 o bo o cr> SVref ected. (After Crary. The travel time of this phase can be deduced from the corresponding group velocity. A theory. which compares well with the thickness obtained by other methods.6 / 0.^P reflected J I \ \ / / N J 20° ^/ (c) 40° 60° 80° 20° Angle of incidence Fig.2 / / N / / X"^ ^ .4 gB X\ J \ / / / \ \ \ \ / J \ \ l 1 ^ \ \ ^ ^SV \ \ . Reflections and refractions at interfaces of a floating ice sheet: (a) P incident (6) SV incident at free surface. 0. A partial derivation and some results are given here. v.PLATKS AND CYLINDERS l.) ft.for the propagation Flexural Waves from an Impulsive Source. (6-54) by setting the phase velocity c = «!. SV incident at ice-water interface.. (d) at free surface.8 / \ -"^^ \ \ \ 1 \ \ \ ^r ^ 1 / / \/ A 0. Square ^P 'reflectedl r\. Consider a plate of infinite extent floating on deep water. Crary found the value 173 H . 6-9. of flexural waves for the case of an impulsive point source located either in air.2 acted 1 \ 1 0) 1 r~~p \ia) / reflected 'b) relative o of •-• d p p root ^^-.' reflected 1 ^SVref 0.water interface. was given by Press and Ewing [65]. \ 0. refracted \ . Equation (6-60) can also be derived from Eq. the thickness being small compared with the wavelengths considered. (c) P incident at ice. with z axis positive upward. where V^ plate. 4-1.2 (6-62) = d^/dr^ + i^/r) d/dr + d^/df. we introduce the velocity potentials ipo and ^2 from which the velocity components q and w and the pressure p can be obtained the water has density as follows: P^=-'^^ The functions <î dipi _ ^^=-^ d(pi ^'-^^ wave equations 1 _ dipi ' ^ = ^'2 c\ ct fa a-i\ (6-61) are solutions of the aS7%. = Jo f QJoOcr)kdk kl. z are used. we can write as formal solutions = of (6-62) and (6-64) for a source at <. Assuming simple harmonic motion (exp {ioit)). ELASTIC WAVES IN LAYEKED MEDIA is an infinite atmosphere having density pi The plate has density pz po. i (6-68) where fNot v^ = Vk' - = 0. solutions of these equations satisfy the We assume now that the boundary conditions for a thin = ^=^= dz dt f-^ dz at^ (6-63) The vertical displacement w^ of the ice sheet satisfies the equation for flexural vibrations of a thin plate: ^ where g is d Wi H piCp 4 d<p2 .2 (6-69) to be confused with the compressional-wave velocity ai. . and sound and longitudinal-wave velocityf and sound velocity 0:2.(kr) ^^ Vo (6-66) Jo Vo <P2 I Q2e''''Joikr)k dk (6-67) Jo w. Following the procedure in Sec. d(po fn_aA\ the gravitational acceleration. =^ at = 0.e--J. V and the relation X = r dr \r \ — dr r dr \ (6-65) dr m is assumed.302 Overlying the plate velocity Cp) ckq. = A (in the air) ^0 = = [" Jo Jo(/cr)e-'"'^-'" ^^ + r Q. Cylindrical coordinates r. respectively. 4-2.t - KT + ^)] (6-72) = where /c —TrVh exp (— exp iUjit ~ ''^ + 7 is a root of the equation G{k) = by Eq. For large values of r. Three simultaneous linear equations can write an expression for point source in the air: result when the solutions (0-fJOj to (6-68) are substituted in Eqs. Cr(/Cj Jo{kr)kdk (6-70) Jo where G(k) - P2 Pi 2 . The residues. direct compressional action of the direct wave. The solution was then expressed as the sum of the residues of the integrands corresponding to the poles given by G(k) = branch lines corresponding to the branch points k and two integrals along = w/a^ and k = oj/ao.j PLATES AND CYLINDERS the factor exp tcot 303 The first term in (0-00) represents the wave emitted by the source. As before. resulting from the being omitted.. (6-72) (6-73) with G{k) defined by (6-71). Integrals of the type (6-70) have been evaluated in Sec. f TT \ 2 H^clk* P2\ Pi/ . (6-66) to (6-68) represent compressional waves in the air and in the hquid and flexural waves in the plate. we the displacement of the plate due to a periodic Wi — 2 Pi ioje 1 ^. The procedure was to transform the path of integration to the complex k plane. 4-2. which diminish as r~*. diminishing as r~^. The steady-state solution represented may be generaHzed for the case of an arbitrary initial disturbance by the Fourier-integral method r/t of Sec. Phase velocity may be obtained as a function of the parameter y = Hf 'ao . Solving for Q. Po Pi 2 (6-71) 1^ The solutions for a point source in the water can be obtained from (6-66) to (6-68) by interchanging the subscripts and 2.-. this procedure leads to the conclusion that the predominant disturbance at a distant point satisfies the relation = U. whereas the branch line integrals represent compressional waves in the media above and below the plate. the remaining terms in Eqs.. (6-63) and (6-04). give the flexural waves.o/O vo/i) exp [i[. the contribution of the residues to the value of the integral (6-70) is approximately 2iruip„KV2 Pi(x/cr/2) 'Pl^^ exp (-. Phaseice. the left of the air source at a c = Q!o- frequency /a = afP/a/H.0 1.0 ^X -^ 0.k dc/dk. po/pi = 0. Let7„ and /„ correspond to c = ao- The portion of the phase-velocity curve for c < ao. Phase. corresponding to propagation at the maximum value of group velocity. pz/pi = 1. a2 = 4.and group-velocity curves are plotted in Fig. the maximum value. largest amplitudes are associated with low-frequency waves. c^ = 11.650 ft/sec.10 Fig. that is. 6-10 for the case ao = 1.500 ft/sec. These waves appear with large . As the frequency (and phase velocity) increases.velocity curve occurring near 7 = 7a. Interchanging the subscripts and 2 in (6-72) enables one to compute the corresponding amplitudes for a source in the water. the steep limb of the group. (6-73).304 ELASTIC WAVES IN LAYERED MEDIA from Eq.00141. and group-velocity curves for air-coupled flexural waves in floating and the corresponding part of the group-velocity curve to maximum would have been obtained had we neglected the effect of the air.06 0. A graph of the steady-state amplitude TF(k) is presented in Fig. since the arrival time of waves of a given frequency corresponds to propagation at the associated group velocity and the wave amplitudes are proportional to W{k). ^^ -— 0. 2.090. However. Study of these curves reveals that a peak amplitude occurs for an for 7 < 7a. and the corresponding group-velocity curve may be obtained using U ^ c -\. corresponding to the phase velocity For a point source in the water. The sequence and character of arrivals at a distant point can be deduced from Figs.070 ft/sec. 6-10 and 6-11. 6-10. For an air shot the first waves to arrive appear at the time t — r/2. and the values t/ = c = ckq for 7 > 7a all represent effects of the air. wave amplitudes decrease and abruptly drop to zero as the frequency /„ is approached. 6-11 for a source in the air.2ao.02 % 0.04 y^ ^^ /^ ^ .08 c 0. steep right branch appear as a constant-frequency train continuing to the time t = r/ao. Cylindrical . where the dispersive waves are predominant.10 7 Fig.0 o o t— 4. 6-10 of arrivals is the same. since the group- appHcable. The constant-frequency train for the air shot and the dispersive train for the water shot are immediately apparent. 6-12. corresponding to the two branches of the groupvelocity curve on either side of the maximum. A water shot is therefore characterized by a train of dispersive waves beginning at the time / = r/2. the amplitudes now follow the curve in Fig. frequency decreases and amplitude increases. In the problems of wave propagation media considered previously.08 J n 0.1 ft thick are presented. in layered Rod in a Vacuum. The constant-frequency waves form the predominant disturbance from an air shot since their frequency /„ frequency. In Fig. 6-13 seismograms from an air shot and water shot for lake ice 1. Steady-state amplitude function for an air source.04 0. Waves propagated according and rapidly decrease in amplitude as the peak value /„. Following this. A special case having some practical interest is that in which the boundaries are cylinders.0 X ^ 2. As time is still increases.06 -^ / 0. However. 6-11.02 L 7a 0. with a frequency close to /„. the cyhndrical rod in a vacuum 6-4. Waves corresponding to the from the frequency decreases to the left branch are dispersive 6. lies close to the peak For a water shot the sequence velocity curve of Fig.0 \^ 0. two wave trains arrive simultaneously. plane or spherical surfaces and interfaces were assumed.2ao.I PLATES AND CYLINDERS 30rj amplitudes with a frequency close to /„. Ref.08 7a 0. but we continue as before to use the general equations of the theory of elasticity. Longitudinal Vibrations. torsional.10 0. lateral (flexural). and These may on be treated in various this well known._d}pld\l/d\l/ dr r dr I d \p a (6-75) dt dz ^ dt For a rod of infinite length and radius a. drical rod immersed in a liquid or More complicated problems concern a embedded in another soUd.12 7 Fig. symmetry and take the The displacements q and w oi a particle perpendicular (r direction) or parallel {z direction) to the axis of the cylinder are expressed in terms of the two functions <p and 1^. 25). For references problem see also Kolsky (Chap. \ 12 \\ \ 8 \\ \ \^ "\ 4 1 1 ^ 0.02 0. 5. Steady-state amplitude function for a water source.06 0.306 ELASTIC WAVES IN LAYERED MEDIA being the simplest example. 6-12. the boundary conditions are that the normal and tangential stresses must vanish on the surface of the cylinder: Vr p^r = atr = a (6-76) . We assume axial equations of motion in cyUndrical coordinates (1-25). as is of vibrations in cylindrical rods are considered here: longitudinal. By (1-26) to (1-29) ^ ^ dr dz I we have (6-74) dr / ^ = ^4'^ dr 2 dz d <p r dr\ dz dr^ r dr 2.04 0. cylin- Three types ways. : S 307 .J2 o « 3 . = —d\p/dr is used in the expressions mentioned above. we = AFir)e' ^P = CG{r)e (u(±l/'2) (6-78) Then by (6-75) we obtain (6-79) dr r dr ^+ If ^ ^+ = k' (fc^ - /')(? = (6-80) we assume of z. (6-75) in the form if is considered. Prz. that v' = v. we obtain the boundary conditions r. 5-3. then the boundary conditions will be independent and 2 kl kl 7^2 (6-81) Equations (6-79) and (6-80) are satisfied by the Bessel functions Jo(kr) and Joik^r). 2 fWhen the cylindrical coordinates are used. a linear elements dr. 'Joihr) (6-82)t Now. dz. Therefore. (6-76) and (6-77). rdx. . respectively. x. ^Sometimes a function In this case one can write symmetry for longitudinal W is W where J\{^) = C'e'"''^'"'J.{k{r) first = —dJQ{^)/d^ the Bessel function of the order. tensions in the direction of the coordinate x are not produced. Azimuthal vibrations were considered in Sec. The stress acting at a free face of such Prr. waves propagating On inserting the expressions (6-83) in Eqs.308 where the ELASTIC WAVES IN LAYERED MEDIA stresses are given by Since wave propagation along the axis of the rod take a particular solution of Eqs. by (6-74) and (6-82). Prx- volume element is built up on the an element forming a Since part of the cylinder surface can have components of axial we consider the case waves propagating in the z direction. we can take A J i Joikr) - iCv i J- Joihr) (u t — vz') (6-83) w for C d ( dJo{k. and only the first two stress components have to be considered.r) T n \ -''^^'^^^^'^-VdrV—drin the positive z direction. (6-89) where o.( 1 - i A:V + If 2v'k\ 1 - I kW ]aki 1 - ~ k'a' ) = (6-87) we omit the factor kâ and neglect all terms of the order a'. we may use the expansion of the Bessel function J./. J.(k.(/cr) + C(2p' - /4) |.u da' = (6-85) dJo(ka) da {2v' - dJo(kâ) kl) da It is difficult to discuss this equation in its general form. (6-84) arid can be given in the form 2m . the root of which gives the velocity of Rayleigh waves at the surface of a solid half space. the longitudinal waves are found to propagate along the cyhnder with the phase velocity CO M (3X P V X + + 2m) (6-88) M where E is Young's modulus. .PLATES AND CYLINDERS in the 309 form 2m -2 J^{kr) - klXJoikr) - 2iCnv ^ J. due to Pochgives hammer [62].r) =0 = at r = a (6-84) 2iuA |. Values of the ratio of phase velocities c/cq as a function of va were .(ka) = I -\ k'a' + ^ k'a' + X • • (6-86) By (kp (6-86) the period equation (6-85) takes the form — 2v )kxa\ 1 — k\a k\\-l /cV + 1 ^ fc. (6-87) reduces to Rajdeigh's equation (2-28). If the radius a of the cylinder is very small (a thin rod).^^^^#^ da 2iv - A:„^XJn(M dMo(k. The second approximation. For va large. Bancroft [1] showed that Eq.= X/2(X -f.(/cr) The period equation follows upon elimination of A and C from Ec^s.m) is Poisson's ratio.a) • 2ifj. In the middle of the fourth is initiated in which the period increases \vith time. the minimum group velocity appears as the prominent train on the last trace of Fig.e. V\ V ^-1.6 2. and c -^ jS at high frequencies.29. In the higher modes c -^ oo at low of 1.) frequencies. (6-87) by Davies [12] for cr = 0. Group velocity obtained from this oscillogram is plotted as a function of period The experimental points are seen to fit the theoretical curve quite accurately on both sides of the The Airy phase corresponding to minimum value of group velocity.0 Fig 6-14. which the period decreases with time.2 of longitudinal ^^ ' 0. 6-16.. 6-14 and 6-15.2 Co \ N. . 6-16.4 (i. 6-1). First three modes.29 as a function a/l. {After Davies. Ohver is [58] has used pulse techniques to study the propagation of waves in long cyhndrical rods. as discussed before. 6-17. In the first mode the long-wav e hmit of phase velocity is determined by Young's modulus. It is interesting to note how closely these results parallel those for the plate (Sec. 3 1.0 08 Ofi -- Co Cr Co V 0. An infinite number modes occur. shown in Fig. and the short-wave hmit is the Rayleighwave velocity.4 1 1.8 c/cq) 1. the first three of which are shown in Figs. Phase velocity waves in a cylindrical rod with Poisson's constant 0. The oscillogram for the first longitudinal mode The first four traces show a dispersive train of waves in trace a train in Fig. Cq = -\/E/ p.310 ELASTIC WAVES IN LAYERED MEDIA computed from Eq. PLATES AND CYLINDERS 1.0 311 0.8 V i'W Co . V ^,'-' 0.6 Co 1 — 0.4 / 0.4 0.8 1.2 1.6 2.0 Fig. 6-15. Group velocity corresponding to phase velocity in Fig. 6-14. (After Davies.) Torsional and Flexural Vibrations. To discuss the latter type of wave propagation in cylindrical rods, all three equations of motion in cylindrical coordinates must be taken into account. Torsional vibrations are characterized by the conditions that q and w vanish and that the displacement independent of xdetermined by only one differential equation. For harmonic waves along the rod axis the displacement v^ becomes proportional to the Bessel function of the first order, and the frequency v^ corresponding to the third cylindrical coordinate (x) is Then the propagation is obtained from one boundary condition Prx =" 0- Iî the first is a rotation of each circular section about its center, and the phase velocity is equal to the shear velocity /3. Dispersion occurs equation is mode the motion in higher modes. direction. bending of a rod or bar can occur in any For flexural waves propagating in a circular rod an axial section can exist such that points vibrating in this plane remain in it during lateral or flexural motion. All three components of the displacement {q, l\, w) are involved in this type of vibration, and we refer to the work of Love (Chap. 1, Ref. 34) where it is shown that for a harmonic oscillation, propagating along the z axis, these components can be expressed in terms of It is obvious that, in general, sin X, cos X and the Bessel function Ji and its derivative. The three arbitrary constants in these equations again are determined by the boundary conditions {prr = 0, p^x = [1]). ^, Vtz obtained (Bancroft = 0), and the frequency equation is thereby This time the frequency equation is remarkable 312 ELASTIC WAVES IN LAYERED MEDIA ^^^'^^^'^^^'tf'Nf^ mmtf ^'^ î^w^Myi^f^ i <^MM"M^f>l^p^^*^^'**^ i t ^m0 ^p u^t^M^ Fig. 6-16. Oscillogram of longitudinal vibrations of a cylindrical rod of hot-rolled steel; thickness 1 in., o- = 0.30, /3i = 10,400 ft/sec. Detector 10 ft from impulsive source. Time marks are 10 /zsec apart. (After Oliver.) in that only one root occurs, as was proved by Hudson [32]. The phase are repro- velocity depends on the ratio of the wavelength to the radius of the cyUnder. Phase- and group-velocity curves for the fiexural mode duced in Fig. 6-18. [58] also is Oliver studied the fiexural waves in rods. An oscillogram of these waves given in Fig. 6-19, and the observed and the theoretical group velocity in Fig. 6-17. The problem of vibrations of a cylindrical rod as well as rods or bars having other cross sections has been treated in different ways {see Love (Chap. 1, Ref. 34), Timoshenko [85]). An approximate equation for fiexural waves was derived and solved by Timoshenko [84], and his results agree well with those obtained from the general theory of wave propagation. addi- To tional extend the results mentioned here to vibrations of a finite rod, boundary conditions at the ends must be taken into account. PLATES AND CYLINDERS 313 1st longitudinal mode o 15,000 g o 10.000 ^n^°-- l-O. ^^D_a . -^ Flexurai ^/ 5000 10 m ode 20 Period in 30 40 50 60 microseconds Fig. 6-17. Elastic-wave dispersion in a long cylindrical rod of hot-rolled steel; 0i = 10,400 ft/sec, a = 0.30, diameter 1 in. (Experimental points by Oliver, theoretical curves by Bancroft and Hudson [1].) n,6 "^ / c -y 0.4 y h/ 1 ' 0.2 0.4 0.8 1.2 1.6 2.0 Fig. 6-18. Phase- and group-velocity curves for flexurai waves in cylindrical rods for a = 0.29. {After Davies.) 314 ELASTIC WAVES IN LAYERED MEDIA Fig. 6-19. Oscillogram of flexural vibrations in a cylindrical rod with as for Fig. 6-16. (After Oliver.) same parameters Torsional motion of the rod will be Only longitudinal and flexural waves propagating unaffected by the liquid. considered. In this problem one uses Eqs. (6-74) along the axis need to be boundary conditions and (6-75) and the 6-5. Cylindrical in a Liquid. (Prr)l Rod = iPrr)o (jP^r)l =0 ?! = ^0 Ot T = tt (6-90) refer to the rod and liquid, respectively. Here the subscripts 1 and For wave propagation in the direction of the z axis the rod potentials (f and yp of the preceding section are applicable. In addition, a potential (Po representing the disturbance in the liquid is needed. The period equation can be derived in the usual manner, and a discussion along the lines of that for a plate in a liquid (Sec. 6-2) can be given. Several investigators have studied this problem, for example, Tamarkin [81] and Faran [16]. The effect of the liquid should be similar to that found in the problem of the plate. the Hquid will be Modes with phase velocity exceeding the speed of sound in damped because of radiation of energy from the rod. Additional modes will be introduced owing to the presence of the liquid. 6-6. Cylindrical Hole in an Infinite Solid. The solution of this problem was investigated by Biot [2]. Denote all the quantities referring to the medium (r > 0) by the subscript i. Then, by Eqs. (1-26), (1-27), (6-76), = yp^ we have (6-77), and W \x djpi Prr a, df + d 2mi (Pi = dr dzJ d (pi at r = a (6-91) aV:" dz' (6-92) dt \dr dz Instead of expressions (6-78) Biot made use of trigonometric functions in order to represent unattenuated waves propagating in the z direction. Changing his notations, (Pi we putf = = AXoimr) cos BSZiikr) sin (vz — ut) (6-93) \pi (vz — cot) (6-94) fSince the z axis differ is usually taken along the rod or hole, the notations in this section in earlier chapters, from those used axis. when the wave propagation was considered along the x PLATES AND CYLINDERS with 315 A and B constant, and m ^Vl-^' k^ = '^ = - (6-95) V fix The phase velocity of waves propagating in the z direction is first c. The modified and 5Ci Bessel functions of the second kind of zero and order 3Co can be approximated by the asymptotic expansion X„(2) ~ i^J e" cos n7r[l + • • •] (6-97) Whittaker and Watson (Chap. 1, Ref. 66, p. 374;]. from (6-97) that (6-93) and (6-94) represent solutions vanishing at infinite distance from the hole. Inserting (6-93) and (6-94) into Eqs. (6-91) and (6-92) and using the equations satisfied by the Bessel functions 5Co and OCi, for large values of z [see It follows Ct3Co(2) & , — nr ( \ 3vi\,2;j "^Q I 1 Ct3Co dz 1 2 "r 1 z dz — uvo ^g_gg^ ^ -^0 dz' _ _ dî ~ - nc J_ v ~ dz ~ Z^'^ ^' we can ehminate 4 A and B and obtain the period equation 2(2 in the form Vi - i J_ 3Cn(â) -^DVl ma (2 -1^ - ^D^JCo(ma) ^^ Vl - ^2 5Ci(?na) If ^g_gg^ Eqs. (6-95) and (6-96) are taken into account, the phase velocity c of the symmetric surface waves becomes a function of the A-ariable va — 2-Ka/l = IT D/l, where I is the wavelength and D the diameter of the hole. If we assume that va —^ <^ that is, the waves are very short compared axial , with the diameter D, and functions 3C„ is if the asymptotic expansion (6-97) of the Bessel used, Eq. (6-99) reduces to Wl - ^1 - V ^^y^^-^ 1 - = (6-100) i; which is the well-known form of Rayleigh's equation (2-28) for surface waves at a plane boundary. Equation (6-99) was solved numerically by Biot, and his phase- and group-velocity curves are shown for various values of Poisson's constant in Figs. 6-20 and 6-21. With increasing 316 1.00 ELASTIC WAVES IN LAYERED MEDIA ^^'^^ 0.98 ,.0^ 0.96 0.94 :^ :^^ 'yy ^^ 0.92 /"/ ^ 0.90 0.88 / 0.5 1.0 1.5 Fig. 6-20. Phase-velocity curves for an Poisson's constant cr. empty cylindrical hole for various values of (After Biot.) 1.00 0.98 0.96 i 0.94 \V v — 0.92 w v^ \v~ V \,^^_ 10 I (7 = 0.50 0.35 0.25 0.90 0.15 0.88 15 20 Fig. 6-21. Group- velocity curves for an empty cylindrical hole. {After Biot.) PLATES AND CYLINDERS 317 wavelength the phase velocity increases from the velocity of Raylei^^h waves to the velocity of shear waves in the solid. The curves terminate at the latter point, which corresponds to a cutoff wavelength l^,„, beyond which prf)pagation cannot occur without attenuation. 6-7. Liquid Cylinder in an Elastic Medium. As an extension of the filled problem considered in the preceding section, the case of a cylindrical hole with a fluid was also investigated by Biot [2]. The radial displacement go in the fluid is now given by 9. = ^ ^jPo \_ (6-101) and the potential <po must ^ 'Po satisfy the equation 1 §}Po d I , (po , . dr r dr dz a^ dt The solution of this equation V'o may be written in the form ior kl = Jo[r{kl - v'fy''-'''' > < v" (6-103) where ka = w/a and ^0 = J.W - klfV'-'''' iovkl v" (6-104) (6-105) since J J<iz) = Iq{z) is the IJ^z) first where modified Bessel function of the kind of zero order. Putting ^ = - (6-106) we obtain the fluid pressure in the form (p..)o = -Po ^- = PocoVo[rKl' - \W'-""' for^ > 1 (6-107) and (p,.)o = Poco'/o[rKl - f)'y"'^-"'^ for^ < 1 (6-108) At the surface of the cylinder the radial components and the normal stresses are continuous, that is, qx of the displacements = §0 (Prr)l = (Prr)o at T = « (6-109) is a composite boundary condition corresponding to the matching of mechanical impedances. The first ratio is determined by Eqs. (6-101), 1 318 (6-103), (6-104), (6-107), ^ if ELASTIC WAVES IN LAYERED MEDIA and (6-108), and we take into account that Ji(w) = -£ Jn{u) h{u) = £ h{u) for^ for^ for^ (6-111) we obtain q q Thus M = -.(f - DVjrKf - iW'-"'' = v{l - f)=/iMl - ^^)^y^-"'> = > < 1 ^^^^^^^ 1 _A^ 'îffrJ'\,,, (1 > < 1 1 (6-113) g/o i' -f)*AMi it for^ (6-114) -f)^ g ?^ In deriving these formulas was assumed that at r = a. The would mean that the liquid is contained in a rigid wrl!. condition g = By Eqs. (6-112), then, the phase velocity would satisfy the equation (f - D'JxM^^ 1 1)^] = (6-115) Hence where ii^ | = or aKf - 1)^ = w„ (6-116) are the roots of the Bessel function Ji. The root ^ = c/olq = 1 corresponds to waves with their plane perpendicular to the axis, while the roots u„ give the dispersion of multiply-reflected conical waves. One can see from Eq. (6-113) that if a liquid cylinder is free at the boundary, that is, (prr)o = at r = a, the multiply-reflected conical waves will be determined by the roots of the equation Jo[m(f For the case account that - 1)*] = elastic (6-117) of a liquid cylinder in an medium we can compute the second ratio in (6-110) from (6-91), (6-93), and (6-94), taking into «The third 1^ - («-^^«) boundary condition (p,,)i is =0 atr = a (6-119) Eq. (6-119) yields the ratio A/B, which can therefore member of Eq. (6-110). This results in the period equation, which after certain transformations takes the form given by Biot: It is easy to see that be eliminated from the right-hand L = Po -^ , . ^t ^^ p, (I' - ,,. 1)^ JolM^' °r):. J,[m(r — - 1)'] (6-120) 1)1 Po/pi." Somers [78] studied this problem for the case of an impulsive ring-shaped line source. Using these relations where a and Eq. where references on previous work can also be found. In this region the waves correspond to those studied in the classical theory of the "water hammer. and for a liquid cyhnder embedded in an infinite liquid. the short-wave hmit of phase velocity coincides with the Stoneley-wave velocity at the interface between two half spaces.23. The group-velocity curves exhibit a minimum value. The short-wavelength limit of phase and group velocity is the speed of sound in the liquid. and the effect of free edges was considered by Love [47]. Two other problems concerning cjdindrical tubes. and various values of po/pi. The propagation of sound waves along liquid cyhnders was also recently considered by Jacobi [35]. were considered in the paper of Jacobi. Equation (6-120) holds for the reflected waves (^ > Ij. which will be discussed in the next section. 6-22 and 6-23. where u = 2ir/l and 2a = D. Xi ~^/3. is Poisson's ratio.PLATES AND CYLINDERS 3] 9 where L denotes an expression equal to the left-hand member of (G-99j. (6-120) and (6-121). 6-8. 6-24 and 6-25) computed from Eq. For ^ < the propagation reduces to Stoneley waves at the liquid-solid interface given by 1 By ^1 (6-95). (6-120). The upper value of phase and group velocity is the speed of shear waves in the solid. one fluid and the other solid.The curves for the first three modes are presented in Figs. ^2 (6-96). or with pressure-release walls. The various modes of free vibrations of an infinitely long cylindrical shell were investigated by Lamb [39]. only three parameters are involved in Eqs. î/ao. Then the variable ^ becomes a function of I'a = x D/l. the cutoff wavelength decreasing with increasing mode number. (6-121). a = j. Biot computed the phase. it and can be expressed in terms of may ^ = ^2 be seen that the parameters c/ao: ^' ~ = ^. 2/xi — ? (6-122) (x\ Sinee ^ + = -^ --^ 2(1 1 a) 2a (e-. The phase-velocity curves for the first two modes were plotted for a liquid cylinder ^\dth rigid walls.and group-velocity curves for multiply-reflected conical waves for the case j8i/ao = 1. With increasing wavelength the phase velocity decreases and becomes practically independent of wavelength {l/D > 5). and (6-106). and a.5. In the Stoneley-wave branch (Figs. Period equations for torsional and longitudinal oscillations of a cylindrical tube were given by . Cylindrical Tube. «h P^ 320 .8 <53. Oi d 00 'C T3 .S o r^ -o -2 ~|Q n -s cr CM . 96 0. Phase-velocity mode with /3i/ao = 1.6 .84 -1.88 ' 0..8 0.4 1.00 0.) .PLATES AND CYLINDERS 1.5. a (i.92 \\ V V v n\ \ s.3 \ 1. Stonelej" = 1/4. {After Biot. 77=0-4 \ 0. e.-- ^^^ Fig.. 6-24. velocity for first-mode phase-velocity curve of Fig.9 0.8 23456789 irD/l . {After Biot.0 === ^-10'l 0.1 w V \ Group ^ = 0.> \ »K^ - 0.0- 10 12 Fig. c/ao) curves for liquid-filled cylindrical hole. 6-22.5 321 1.2 \ «0 1.) 1..4 1. 6-23. 6 0. pp. Three boundary conditions must be satisfied. Group velocity for phase-velocity curves of Fig.322 1.8 1. we assume for the first of these pi.88 0.84 ^/ ^/ ^ ^k> ^ I 0. A solution for this problem can be written in the form ^0=2 <Pi A. 1. and 323) and Lamb [40] for early investigations of this problem}. we can make use of velocity potentials similar to Eq. Ai„. for n = 0. 6-25.80 Fig.92 U_ 0. the propagation of waves in a was discussed in several papers {see Rayleigh (Chap. II.) Ghosh [22].0 0.00 ELASTIC WAVES IN LAYERED MEDIA 1 ^ = 0. A case with practical interest. (6-103). and therefore two approximations are made. We can consider hquid-filled tube.4 0. B^„ are arbitrary constants and m = The symbol it is '\/klo — v^ 1 = VA. 45. that is. 6-24. (After Biot. Again considering solutions with axial symmetry. a^ for the wall problems that the density and sound velocity are and po. Following Jacobi [35].96 Po 0.r)y (nx+Vz-at) (6-125) where A„. H'^^^ where the argument imaginary. expressing continuity . i/l". 158 either a hquid cylinder with a liquid wall or a liquid cylinder with a thin solid wall. In this problem the yielding of the walls cannot be neglected. vol.„N„{m. Hankel function best to use For convenience is in computation. Ref. A relatively simple form can be obtained for modes independent of the angle x. ao for the cylinder.L — v' (6-126) A^„ denotes either a Bessel function of the second kind or a of the first kind.JJmr)e' (nx+Vz—oit) (6-124) = H [Ai„J„(mir) + B. Engrs. 1952. 35. Aeronaut. : 6. (6-128) separately and locating intersection points. 588-593. 8. we obtain PoAoJoima) — pjAu. pp. Mech. 1951. Union. 1941. Bancroft. L. J. vol. A. and two different cases cto ^ cxi were discussed. Crary.: Propagation of Elastic Waves in in a Cylindrical Bore Containing a vol. Waves Beams.oJo(wia. Trans.(mâ) mJ'ima) Jo{md) p^mi JoimfOjAûimta^) pi J (^{mâ)N n{mia^) — J„{miai)No(mia) C6-128) The derivatives of the Bessel functions of zero order can be replaced by first functions of the order. vol. Christopherson.Ji<(inid) — — p^B^„N<. Appl. vol. 3. D. Proc. Bishop.{mâ) A. 1947. 1889. Cambridge Phil. Appl. M. L. Biot. Quart. 165. Brown. second case it as the frequency increases. Jacobi [35]. vol. pp. InM. A?n... vanishing and of the pressure at the outer boundary of the tube.. 1952. E. 997-1005. P. using the formulas Jo is = —Ji. 243-247.) + — Bt. vol. 293-300. by a metal. approaches the sound velocity of the outer medium Wave propagation in a liquid cylinder enclosed [26]. pp. R. vol. J. 59.: The Propagation of Elastic Waves in a Rod.N'. 15. tti. Akust. 81-10-4. 1-22. D.. Mech. C: The Equations of an Isotropic Elastic SoKd in Polar and Cylindrical Coordinates. of radial components of velocity at the inner boundary. Phil.. The Velocity of Longitudinal Waves in Cylindrical Bars. Aô = — A''!. E. pp. approximate expression for the right-hand member was used by Jacobi. 1954.PLATES AND CYLINDERS of pressure r r 323 = = a. pp. Z.: Seismic Studies on Fletcher's Ice Island. Rev. pp. 1948. 23. Clark. T-3.â^) The period equation may now be written in the form J.. S. and Since only the terms n = are used in (0-124j and fG-125j. . Phys. Their Solutions and Applications. pp. 7..: Theory of Sound Damping by Thin Walls. Gronwall Fay.: vol. Mag.: Longitudinal 293.>NJm. The of graphical solution of this equation obtained by plotting the curves An each member of Eq. Soc. 38. 2. and Fortier [17]. and Fay [19]. Effect of Shear on Transverse Impact on Beams. {London): Appl. pp. D. G. 250-369. A. Mechanics. 5. In the first case the phase velocity approaches the sound velocity of the inner in the medium. Trans. 9.walled tube was investigated by Lamb [40].(mâ) = = = (G-127) mAnJoima) — miAuJnimiO) m^Bu. D. Duwez: Discussion of the Forces Acting in Tension Impact Tests of Materials. Phys. Chree. J. 14. Fluid. B. Cremer. 176-184. 7. 3. 1942. Geophys. and P. vol.t(mâi)N!. Cooper. REFERENCES 1. pp. 280- 4. 25.: A Critical Study of the Hopkinson Pressure Bar. Sound Scattering by SoHd Cylinders and Spheres. R. Amer. 27. F. Rev. 1948. D. Amer. J. New York.. S. Fay. . Congr. 1927. S. 459-462. E. Bohnenblust: The Behavior of Long Beams under Impact Loading. Z. Rev. Heins. and M.: The Longitudinal Vibrations of a Liquid Contained in a Tube with Elastic Walls. u. Akust. Math. pp. 63. Roy. 16. vol. 1947.S. D. 145-168.. 6.. E. Ghosh. G. of Elastic 215-220. pp. Natl. J.. D. L. 23. 13. J. K. 14.: vol. /. pp. 11. Gronwall. Appl. 1951. 1948. Fay. Appl. J. P. 1923-1924. Fay. Clark. R. Juhasz. 1946. 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Properties of a Pair of Semi-infinite Parallel Plates: Quart. E. 248. II. A. Ubertr." Columbia University. Akust. 1952. Soc. elek. Z. Rev.. A. Mech. 405-418. D. 5.. E.S. 31-40. Quart. 30. 1950. 54. vol. 18. Davies..: On the Propagation of Ultrasonic Waves in Wires. 21. 119.: Dispersion vol. 1948. pp. Publ. and O. 46-51. Blechschmidt: Experimentelle vol.. 217. M. vol. 1952. SHsm. Czerlinsky. vol. and E. 7. J. First U. 618-626. Hudson. 1943. pp. 1942. Hartig. Fortier. 22. /. vol. Physik. Gogoladze. p. pp. J. vol. 1949. pp.R. pp. Soc.: Graphical Analj^sis of Impact of Bars Stressed above the Elastic Range. 27-34. 1947. 25. 23. 24. American Society of Mechanical Engineers. (London). Trans. D. Faran. 26. 157-166. 4. Fliigge. 313-329. II. Springer-Verlag OHG. Acoust. in 18. J... 1943. Duwez. vol. Depart- ment 31. H. 6. 1922-1923. G. Mechanics. Soc. 1953. and O. Acoust. Acoust. and C. 19. 17. pp. Appl. pp. pp. Ann. angew. vol. 339-346. V.. Math. Inst. 15. vol.. 15-48 and 113-142. F. U. p. ELASTIC WAVES IN LAYEKED MEDIA : Cremer. pp. 1948. 1947 (in Russian). /. Soc. 8.324 10.: "Forced Motions Rods. 136-139." vol. Acoust. L. Die Ausbreitung von Biegungswellen : in Staben. 13. Set. Giebe. 1953. Herrmann. fiber 329. pp. Amer. part I. E. 30.. Soc. 1933. pp. 71-83.. E. A. Math. vol. und theoretische Untersuchungen I Dehnungseigenschwingungen von Staben und Rohren: 417-485. E. R. M. Calcutta Math. Swanson: "Transverse" Acoustic Waves in Rigid Tubes. vol. pp. Soc. of Civil Engineering. 23. J. 12-17. 1942. 375-457. vol. pp. V. 2. pp. vol. Carlson: The Reflection of an Electromagnetic Plane Wave by an Infinite Set of Plates: I.. Holden. 27-38. 1928. A. Physik. Franklin Inst. Arch. vol. H. R.: Waves in Liquid-filled Cj'linders. R. Acoust.. 19.. BerHn. vol. Phil. pp.. 956-969. 240.: Longitudinal Modes of Elastic Waves in Isotropic Cylinders and Slabs. 24. R. G. 30. 39. 43. Plastic Wave Propagation in a Bar of Material Exhibiting a Strain 48. pp. Wolf: Plastic-wave Propagation Effects and J.. vol. Malvern. H. 1950.: Transmission Reverberant Sound through Double Walls. 18. vol. vol." J. 55-78. Soc. Prikl. Z. London. Rev. Acrmiit. pp. vol.John Wiley Soc. 383-406. 32o 34. E. Akad. & Sons. 379-386. Appl.: : Rate 50. L.: On the Flexure pp. and H. L. Mindlin. 1889. A. Frey: "Fundamentals 19. Lazutkin. Mechanics. . 1552-1556. Phys. Akad. L. 21." Columbia University'. 1951. 1952 (in Russian). pp. Meh. /.S. Mindlin. Soc. Liquids by J.. vol. pp. 1951. Appl. D. Prikl. Levine. Russian). A.S. Lebedev. vol. pp." . 20 -127. Roy. 1898. 1951 (in 42. 37. 52.. 1951. Mechanics. 52. pp. Meh. 1889. D. H. vol. D. New York.S.R. 1951. Iliiter. vol. E. 1952. H. N. 1946 (in Russian). 1951. Propagation of Waves over the Surface of a Circular Cylinder of Compt. Mat. London Math. 44. 1949.Math. Rend. Phil. Inc. 31-38. pp.: Infinite Length. vol. Department of Civil Engineering Rept. 53-A-42. Trans. lar Pipe.: Propagation of Elastic-plastic Waves along a Cjdindrical Rod.S 3. 13. 270-279. 18. 274-2SS. pp. pp..53.: Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic Elastic Plates. IL: On the Velocity of Sound in a Tube as Aff(!cted by the Elasticity of the Walls. Appl. Acad. L. Prikl. J. : On the Small Free Vibrations and Deformations of a Thin Elastic Shell. Kane. 355-365. F. T. Soc. Love. Kinsler. U.PLATES AND CVLINDKK. Meh. New York. 47. Acoust.. 55. R. Appl.: "Rcdeelion of Fi(!xural Waves at the Edge of a Platf. 405-411. Mat. Effect. 53. Lunts. (London). H.S.. pp. pp. Nauk S. in High-speed Testing. J. 1889. 119-146.. Soc. and R. angevj.: On the Propagation of Unloading Waves in the Case of Linear Hardening. (London). pp. A.: On the Deformation pp. J. vol. 35. pp.. 18./. Ponclroin. E.R. A. Schwinger: of On the Radiation of Sound from an Unflanged Circu. 93. 24.. 94-100. of an of IClastio Plate. F. Herrmann: Waves in . S. On the Propagation of Spherical Waves in an Elastic-plastic ^Medium.. McSkimin. Soc. A. Nauk S. S. 38. Acoust. H. 1. W. Sci. 491-456.S. 54. R. J.: t)ber die Fortleiturig von UltraHchallvvcIleii in U'mU'M Stiiben. D. H. 16. E. A. R. vol. 1949 (in Ru. : pp. Mat. Proc. and H.R. 40. T. of Malvern.: On Waves in an Elastic Plate. vol.^./.s Along Lifjuid Cylinders.. Lamb. 1952. L. and G.S.. vol. Measurement of Dynamic Shear Viscositj' and Stiffness of Viscous Means of Travelling Torsional Waves. 1949.. 1949..50. Phys. Akad. Amer.R. Amer.ssian).. 203-208. 36. Amer. 114-128. Appl. 49.. Proc.: The Propagation Longitudinal Waves of Plastic Deformation in a Bar of Material Exhibiting a Strain-rate Effect. Department of Civil Engineering Rept. 70-90. Rev. H. Lee. Proc Manchester Lit. L. 45. Jacobi. 41. 21. vol. 46. Proc. H. Propagation of Sound Wav(!. Ih. Nauk S. Lamb. vol. vol. 625-628.: J W. 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K. and Oliver: Model Study of Air-coupled Surface Waves. 1938. Oliver. Rakhmatulin. pp. Soc. Natursforsch. p. K. vol. Meh. Hart: Transmission. The Transient Motion of Sound Waves in Tubes. 302-306. S. vol. Schoch. vol.326 56. pp. D.. Morse. 9. Acoust. : ser. F. 5. 61. 324-336. vol. Acoust.. Bull. J. 27. 1950. 369-374. pp. K. pp. M. Trans.: Schallbrechung und Schallbeugung.. 20. . and G. Mat. A. Acoust. vol. 1953. pp. 72.. 1951. 223-261. and J. p. J. 57. 12. 1955. : Dispersion of Elastic Waves in a Cylindrical Rod by Wide-band Short- duration Pulse Technique. Mag. Acoust. Meh. F. 249.: Propagation of a Wave of Unloading.S. Press. 217-219. Osborne. 1951. F. pp. pp.S. 11. 313-335. 32. 43-46. W.. vol. 1945. Y. vol. pp. Sheet.. 58. Bull. Mat. 1889. 12. 11. p. 673-678. Soc. pp. vol.. pp. 39-46. reine angew. diinner Flatten. Pearson. Prikl. H. Math. vol. A. Meh. and S.: Elastic Waves and Vibrations of Thin Rods. : in Rods of Rectangular Crossa Section. Soc.. 22. 91-100. 25. Rakhmatulin. Lord: On the Free Vibrations of an Infinite Plate of Homogeneous Iso- tropic Elastic Matter.. 3a.. 833-838. Nishimura: Rayleigh-tj'pe Waves Propagated Along an Inner of a Body. Rakhmatulin. 65. Shapiro: On the Propagation of Plane Elastic-plastic Waves. 3. Prikl. F. IX. vol. Phil. Acoust. Acta. 1948 (in 70. 85-92. 65-74. Nauk S. Rinehart: Surface Motion Associated with Obliquely Incident Elastic Waves. Russian). 1948. J..S. Mat. J. Amer. pp. vol. Schallreflexion. tJber die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiszylinder.. pp. vol. vol.R.. 1876.S. F. Math.R.: Investigation of the Laws of Propagation of Plane Elasticplastic Waves in a Medium with a Variable Elastic Limit. vol. Pochhammer. Phys. Prikl. Phys. Study on Surface Waves. 76.. vol. Press. vol. Helv.. D. and Guiding of an Exponential Pulse by a Steel Plate in Water: I. Akad. Amer. 75. 1938. or Scientific Papers. {Tokyo). 1950. /. 17. Soc. 81. Reflection. 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Geophys. R. The Propagation of Torsional Plastic Waves in Circular Tubes and Shafts.i. pp.section.. pp. M. Rev. pp. : H. Mat. 1947. 89. G. Y. P. 1952. J. 90. : 93. Timoshenko. Uflyand. Mechanics. Phyn. pp. 41. vol. 33. Appl. Math. New York. Timoshenko. 329-343. P.. Wood. C. : "Radio Engineers' Handbook. Acoust. C: The Intrinsic Inelasticity of Large Plates. 12. S. 94. Mechanics. 1st ed. and L. Div. Van Xo.. Surrounded by an Elastic Solid. Tamarkin. 19o. 87... Prikl. Soc. 1950. in Solids. Ann. M. 59. Rept.: On the Impact Behavior of a Material with a Yield Point.: "Vibration Problems in Engineering. S. IX. Proc. 1937.'i. 614-622. Amer. 91. 92. 39-54. pp. Inc. Mechanics. 88. Congr. 85. 1949. 83. 1948. 81. Brown Univ. 1949. Stanford. Zener.. D. Appl. D. S. P. 1948 (in P. Mechanics. 612-616. V. 744-740. P.: The Propagation of Waves in the Transverse Vibrations of Bars and Akad.. pp. Physik: FolgeH. 7 (suppl. Tech. vol. Trauter. 1948. pp. Stenzel. vol. pp. pp.. .'i40. McGraw-Hill Book Com- pany. Plates. 337-341. 16. Wolf.. Appl. 19. Griffis: The Propagation of Plasticity in Uniaxial Compression. S." 2d ed. Terman. A Contribution on the Velocity of Longitudinal Eia. and L. White. 287-300." York. vol. 1949. 24. Appl. 327 Somers. /./. J. 43. P. olo-o21. vol. E. Phil. 256-260. M. vol. of the Laplace 86..: On the Transverse Vibrations of Bars of Uniform Cros. 84. 1922. Mechanics. Russian). 1941. Nauk S. pp. pp. /. vol..: Propagation of Acoustic Wavos in a Liquifi-fillod Cylindrical lIol«. 1-31. All-50. vol. vol. Mag.: Scattering of an Underwater Ultrasonic Beam from Liquid Cylindrical Obstacles. Inc.R. vol. ser. Niiovo cimenlo. 1943./. M. 1921. Appl.S. Seventh Intern. New S. 1. P. 1942. Grad. pp.S.: The Dynamic Stress-strain Relation of a Metal with a Well-defined Yield Point.: Das Schallfeld eines Strahlers in einer Mediumschicht mit schallweicher und schallharter Begrenzung. vol. Meh. 1943. i'. 15. vol. .. P.: The AppHcation Transformation to a Problem on Elastic Vibrations. Appl.. White. /. E. E.s-. Griffis: The Permanent Strain in a Uniform Bar Due to Longi- tudinal Impact. Company. F. 82.: On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars. 80. Appl. Mag. 79. vol. White.stic VibrationH in Cylindrical Rods. 125-131. pp.. 521-525.strand Timoshenko. 14. Phil. H. pp. Phil. Mag.: On Longitudinal Plane Waves of Elastic-plastic Strain .. 21. vol. 669- 673. Phys. IW2-.PLATES AND CYLINDERS 78. White. CHAPTER 7 WAVE PROPAGATION IN MEDU WITH VARIABLE VELOCITY 7-1. and two similar equations are given by for the other components. motion (1-7) and (1-1 1) in which y. Thus we have P -:ra dt and p are now functions of = pX -+ —— + ~ dy dx H -r— dz U-l. Wave Propagation in Heterogeneous Isotropic Media. where the stresses and . dy + . z. /dw "^ du dz dx dx ^ dy [dy dx) dz \dx (7-4) "^ "^ dx \dy dx) 328 dy dy~^ dz \dz dy . ix.du dji (du dv\ djj. "^ cf djj. dw T~ dz Inserting (7-2) in (7-1) we obtain M p—-2 df = pX + — dx {\d) + mV d/i + a?/ M 7- dx dw "^ d. the equations of motion take the form .T 5m /5w ^\ a. Problems kind are mathematically more difficult than those considered since each investigation must begin with the most general equations X.T/ 5/i /dio '^ du\ dzl "^ dx Uy ~^ C7_o^ ^ dz \dx ^ After a simple transformation. of X. „ e T = div s = du r dx dv h T" . of this earlier. dji djj. (7-7) and (7-8) we du dt dX dx (7-9) . iv w = must 2 (7-8) dz satisfy equations such as From Eqs.V'»-. dvA dyl . /dw du\ dz) . for fluid media we put /x = 0. the equations needed for various heterogeneous media would follow from Eq. (7-4) that is. Obviously. (7-6) through a specialization of conditions. is if the sum understood to represent the double scalar product of the gradient of rigidity V/x and the symmetrical strain tensor .+.|f-2| (7-4) diJ.) dx \dx dy \dz It is easy to see that these equations can be written in vector form. ' V/x du ' \dx dy dz) and du dx \(du O 2\dy^ i(du 2\dz dv\ l/du 2\dy dv\ "^ l/du 2\dz l/5l^ dx) ^ dw dx dv dw dy (7-5) dx) dvo" dy \(dv 2\dz ^ dw\ "^ dw dz in dx) for 2\dz dy) Then the vector equation is wave propagation a heterogeneous medium = p^ dt pF + V[(X of + 2m)0] + mV's nVd - 20 Vm + 2(Vm-*) (7-6) The equation apparently given first wave propagation in a form equivalent to (7-6) was by Uller [75] and independently by Yosiyama [81]. of the three last terms in each of Eqs. for example..WAVE PROPAGATION IN MEDIA WITH VARIABLE VELfJCITY 329 '>^=^^ + |l<^ + 2. v. we have v(\9) (p p^ = pi = evx + xv< (7-7) Now we introduce the potential defined d<p bj^ d(p u = d(p dx dy see that u. Assuming also that there are no external forces acting. (In ((Jv . F = 0. n^dw dz dz fcont. Thus.). y. If the relative variation of p. (1-51) for wave propagation in heterogeneous media. z) and /3 = I3{x. 1).y.. Eqs. that if -^lf=« p(x. 7-2.. Y.z)V'<p = vX/p. z).) +l fW- /'(P. {I -\- az -{- bz 2. m. 555]) ^=i if + ^'» +Y. coupling between these two wave types occurs at every point of the medium. ""^ . = where a aXx. then. Otherwise.. + Z. = c^- The motion if /'(po) = yPo/po Then the potential satisfies the equation ^ = eVV + (x|+K^. Po = /(po)irrotational in the case of "convective" equilibrium. i. either as was shown by Sobolev a or = = az (7-15) « . Sound Propagation in a Fluid Half Space.) \e. z) is a (^-10) constant the displacement potential for a heterogeneous isotropic fluid satisfies the wave equation (7-11) ^. and it is not possible to make a clear distinction between them (see also Chap. y. + z|) It (7-14) hoff's [69-71] that it is possible to generahze Kirchformula Eq. y. representing variations of d_ from the usual wave equation by terms Equation (7-9) can be written in the form dx It is seen.-- (7-13) the forces X.e. is Z are constant and c^ = ypo/po.. (7-4) may be approximated by two wave equations for compressional and distortional waves with a = a(x. )' (7-16) .330 ELASTIC WAVES IN LAYERED MEDIA d where = VV> w'hich differ X. § = aW ot in h<p (7-12) which the last term represents the effect of heterogeneity of the is medium. and X is very small over a wavelength. p. The small motion of a gas about a state of equilibrium determined in general by the following equations (»'" (see Lamb [44. The effect of variations in density was investigated by Bergmann Yosiyama [81] used an equation of the type [9]. . Pekeris [58] considered the problem in which the velocity in a fluid depends on the depth z. 2). vanishes as before. It is assumed that the density variation is sufficiently small to permit the use of Eq. an expression A{k)J. 4-2. Notations for sound propagation from a source at (0. represents a wave propagating from the source located at Free surface Fig.WAVE PROPAGATION IN MEDIA WITH In the first VAHIAI5LE VE1/)CITY . z = If z. and. (7-11): 2 I d (f a (7-17. If we write 2zk = x and Fiz) = F{x/2k) = W{x). Eq. 7-1.V}] case the velocity decreases with depth above the plane 2 = shown in Fig. the pressure. z) to a receiver at (r. (7-20) is reduced to the form ^+ dx^ f^l _ \a^x^ 1 )TF 4/ = (7-21) . both components of velocity are continuous at the level 2 = 2.{kr)F{z) (7-18) <p = will satisfy the equation VV + provided that the factor F{z) is kl^ = (7-19) an integral of IP + az' A 4-. _ A \a'z' I = (7-20) where the velocity distribution given by Eq. the time factor exp {iwt) is omitted. hence the potential (p. 7-1. as was assumed in Sec. (7-15) is already taken into account. at At the free surface z ip = Zq. except at the source. and = AlUkzo) Thus we can put F. (7-20) can be written as follows: F = AzÎinikz) + Bz'KUkz) kind (7-26) where / is the Bessel function of the and 3C is the modified Bessel function Kn(z) first mth imaginary argument = \^j cosmrWo. = F. Eq. where G{x) = G(z) dV dx^ I 1 - 1 I G = X dx n = (-. (7-26) are of the order in. On the other hand. together with the is forms the fundamental Eq. 360)] 1 4 1 2 - n 2 = CO -2 a first. = F CzÎUkz) in (7-28) z The potential we obtain (p and therefore Eq.: 332 ELASTIC WAVES IN LAYERED MEDIA satisfies this Whittaker's function Wo. for the + BKUkzo) the source (7-29) medium above = Bz^ XUkz) . the substitution F{z) = z^Giz) (7-23) yields the equation (TG dz^ If ^ IdG z dz'^ .n{2z) (7-27) Both functions in Eq.n{x) equation if we put [Whittaker and Watson (Chap. 66.„ corresponds to a wave propagating upward.lUkz) lUkzo)^ (7-30) Now the constants B and C should be expressed in terms of the intensity of the point source. Since 3C. (7-24) takes the usual form of a Bessel equation. This result will be obtained as in Sec. (7-21) TF^ „(-x). 4-1 by imposing the conditions F. Ref. (7-26) vanish at = Zo. p.a' 4/ G = z' (7-24) we change dG the notations as follows and put kz = ?x. dz dz = -Dk at z (7-31) . we write for the medium below the source F. 1.-\] ^"^ \a 4/ (7-25) a Now the solution of Eq. (7-22) The second system of solution which. 3C. where it becomes infinite in such a way that its integral over the plane is finite.liM) lUkz) Kjkz. Thus we have dtp.„(x) ~ i^^j e-' lUx) . (7-37) and (7-38). WAVE PROPAGATION IN MEDIA WITH VARIABLE VELOCITY z 333 Then (p is continuous across the plane = z.{2TxyV (7-41) .Jikz) - lUkz) L ^2 ^HH dk (7-39) iinKkZQ)_ and = r Jo Jo(kr)C(k)z'IUkz) dk (7-40) where B{k) and C{k) are given by Eqs.. This satisfies the condition for a point source. Pekeris also derived an expression for the potential for a whole space with the same velocity gradient.. if we integrate with becomes prodk which vanishes everywhere except at the source.) - -D + Ct'IUkz) (7-33) From these two equations we obtain B and == ~IUkz) Dz-'^ 3C. d<p\/dz — portional to Jo 'J(){kr)k B Bi- azUkz) .„{kz)kz = Dkz'IUkz) ^••(^^) (7-37) Dkz' """^ . This expression is readily obtained b\' assuming Zo -^ ^ Taking the asymptotic values of Bessel functions . and.„(A:2) - X[Skz) (7-34) llinikz) C == Dr' \lUkz) UUkz) 5ZUkz) - SZUkz) L lUkz) lUkzo)^ (7-35) When we make use of the relationship rn{x)Kn{x) - In(.tit) (7-38) potential <p is determined for both regions above and below the to oo we integrate with respect to k from ^1 = f Jo Jo{kr)B{k)z^:yi.X)3Zn{z) = 1 (7-36) these expressions take the form B = C = Thus the source if Dr''I.Jbz respect to h between the limits and oo.) Ln(kz<n = Cljkz) (7-32) KU'kz) - SZUkzo) IJkz. the procedure used in Chaps.zf R\ = r^ + (I + zf " (7-44) we obtain r Jo{kr)Ukz)Kr. He obtained the solution as a sum of residues in the form of a Fourier-Bessel series in which Bessel functions of imaginary order and complex argument are involved. From the physical viewpoint. p. „_c L Iin{kZo)j ^HN"! = 2 (7-42) we have <P2 for < f Jo < I = Dz¥ Jo{kr)IUkz)Ki„ikz)k dk (7-43) The last integral can be replaced by a simple expression (Fig. can penetrate into this zone. nRi z 2. this problem is of interest because of the occurrence of a shadow zone into which no rays penetrate. and the intensity can be derived from the residues of Eq. (7-47) for a point source of unit strength D = For vanishing-velocity gradient. (7-47) goes into the familiar form for a spherical wave in a homogeneous infinite medium where the time factor is again introduced. n/z —^ka..334 ELASTIC WAVES IN LAYERED MEDIA we obtain lim and. disregarding only the viscosity and the heat conduction of the medium. if we put z = z. (7-46) becomes ^^ of the source. wave propagation in an inhomogeneous and moving medium from a more general viewpoint.{kz)k dk Jo = -^ rCiUo { \/t2 ^' + ^) — Ki/ (7-45) Thus the solution (7-43) for a point source in an infinite medium with constant-velocity gradient can be written in the form ^2 = Dz'z^ (R2 \ii2 K1K2 \p WW + fii Wl — ^1/ V'" ^ 1I1K2 WW^ Dz'Z^ -2inv (7-46) /« . They are derived from the equations of hydrodynamics. 297). . however. Pekeris also outHned the theory for the case of a medium with variable-velocity gradient. 7-1) (see Pekeris [58]. Even in the case Blokhintzev [10] established the equations of . For the vicinity R2 = 2z. Pekeris evaluated the integral (7-40) by transforming the path of integration into the complex k plane. (7-40). 3 and 4. Diffracted radiation.a\ where tanh v = R^/Ri. Eq.2^ exp D . and Eq. therefore. If we put Rl =r' -{-{z. However. In earher papers as well as in the paper above he applied the method of normal modes and was able to present the solution in the form of a discrete spectrum combined with a continuous one. it was shown by Blokhintzev that for hquids this mon. instead of Eq. 7-2 (Ewing and Worzel [25]). see Dyk and Swainson [21] and Anderson [5]. the assumption that the sound-velocity gradient approaches zero in the uppermost layer of ocean can then explain many features of the shadow zone.WAVE PROPAGATION IN MEDIA WITH VARIABLE VELOCITY of constant entropy 335 and no motion in the medium equation satisfied by the sound potential is other than waves. This equation is of the Schrodinger type. In Morse's opinion. SOFAR Propagation. First. Special cases of wave propagation in heterogeneous media can have important practical applications. and the solution can be expressed in terms of Hermite polynomials. however.(l ao{l - az') bz) ^ a„(l + 2azY^ ^ ao(l + 26^)"* ^^ ior H' > z > (7-4S. This variation in sound velocity is due to two factors. a more general equation. The problem similar to that treated in this of section but concerning the propagation electromagnetic waves was considered by Krasnushkin cited [41]. since the case of the linear decrease of sound velocity with depth discussed by Pekeris does not agree with actual measurement of underwater sound propagation near the sea surface. He assumed that a vertical electrical current produces electromagnetic waves in a nonmagnetic medium having its dielectric constant a function of z and obtained. general equation can be approximately replaced by the usual wave equation. the sound velocity decreases to a minimum at 400 to 700 fathoms but increases from that depth to the bottom achieving a sUghtly higher velocity at the bottom than at the surface. The mean-velocity-depth curve in the Atlantic Ocean is presented in Fig. another law should be used. the velocity either increasing many parts of the deep ocean. long-distance propagation of sound waves in the atmosphere and in the oceans is possible because of the favorable variation of sound \'elocity with depth in these layers. It was pointed out by Morse [54] that. In For related studies in the Pacific Ocean. or decreasing with depth. He expressed the sound velocity in terms of depth as follows: a a where = = -I a. (7-20). In most of the cases considered in this chapter simple distributions of velocity are assumed. the temperature in the oceans decreases rapidlj' -îth depth to a little above 0°C at about 700 fathoms in the Atlantic and 500 fathoms . In particular. (7-49) for 2 > H' 10"^ ft- = ^ = WO^TW) ^ = (7-50) and H' is the depth at which the law of velocity variation is changed. the complicated. ^^\ V^ A i H? Oil lOC co'^jf/ ( - V* r 'nl i î^ ^ k: 7 ^r ^ ^ H \iX /\ .d9a 336 .— — — ^ 0019 0505 4^ ooos -se. ' ^ ^ 8 *> 055t7 00617 05817 ^ C C O ra (/) 03 .^^r LfS cn\ iri 1 o ° / / o o o —— 1 1 1 \ — 1 1 1 — ———— 1 1 1 J 1 1 o o o CM Pm SLuoyje^ uj Lj.ô _ _«— ^^— —"^ /^ ^^ A/ \y V// S^^ 1 / i^ 1 "^ .. Part of the initial sound waves are the initial angle lies between sorption. In this and are bent way. The name "sound channel" has been used for this type of velocity structure. the representation by rays is sufficient to account for much of the character of SOFAR signals. However. Ewing and Leet velocity of propagation varies continuously in we assume that the a medium. Two-dimensional spreading. In a case such as this. the unique character of the SOFAR signal. A limited development of the wave theory was attempted in different . especially the abrupt termination which allows the arrival time to be read with an accuracy better than 0. ab- by an obstacle alone limits the horizontal range of propagation in such a case. 4-4).WAVE PROPAGATION in the Pacific. 7-2. WTien such a theory becomes available it may explain the puzzHng features in this period range on seismograms of earthquakes with large oceanic paths (see Sec. Rays with an inclination SOFAR. IN MEDIA WITH VARIABLE VELOCITY 337 and ihio v(3lo'-ity decreases in this range as a consequence. or long-distance.05 sec. If the inclination exceeds 15° at the axis the rays must be reflected at both the surface and the bottom and are called reflected sounds. Below these depths the temperature decreases slowly to the bottom. rays repeatedly return to the depth of minimum velocity and cross it in the opposite direction. If is a linear function of (see. They have been called RSR (refracted and surface-reflected) sounds. and a bent upward. the sound waves is follow curved ray paths because of refraction. a complete wave theory of propagation in the SOFAR channel must consider the elasticity of the bottom as well as the heterogeneity of the water. Propagation of the sound initiated by a source of spherical waves located minimum velocity has characteristics which may be explained by Fig. between 12 and 15° to the axis of the velocity minimum are also curved but return to the axis after reflection at the sea surface. and (3) sound duration depending upon distance in such a way that the distance may be estimated from the signal duration at a single station wdth a precision of about 3 per cent. signaling in the oceans displays the following (1) characteristics: the extremely long-range (2) transmission of sounds (probably 10. and the increase in velocity in this part of the ocean is caused by the effect of increasing hydrostatic pressure on the compressibility. or interception therefore confined to a channel. Such a theory is necessary for waves mth periods between 1 and 15 sec. The rays in this figure were calculated from the mean-velocity-depth curve under the assumption that the water column at the depth of is composed depth of six layers in each of which the velocity [23]). for example. a ray starting with a small angle above the horizontal ray which starts at an angle below the horizontal is bent downward.000 miles from small bombs). With the high-frequency-sensitive detectors used in underwater acoustics. For oceans the rays behave in this manner if and 12°. a weak disturbance is arriving. and in an elementary way he derived some formulas for the effect of this velocity gradient. (I -\. i — L A B 0) a: sz. 127). The depth. In a second paper [12] on the subject Brekhovskikh suggested an approximation of the velocity-depth curve ADE by a broken hne ABODE (Fig.) sity at the principal part. "2 a (z) Sound velocity (&) Fig. Ref. He applied Rayleigh's consideration of rectilinear waves incident on a curved boundary to the case of a plane boundary and curved rays.) end of a signal decreases as r~'. 25. (After Brekhovskikh. (See also the discussion at the end of Ref. 1. These waves are called by Ewing and Worzel the RSR waves (see Ref. Fig.az). the coefficient in the expression for the velocity a. p. 546.) (b) Assumed velocity-altitude relationship for atmosphere. 7-3. is H is Sound velocity . which is due to reflections from the bottom. 2. 16. A more extended discussion of the problem of a sound channel was given by the same investigator in other papers [14-16]. vol. At large distances the maximum of sound inten- factor a. oa. 7-3a). (After Haskell. z being the variable depth. Following the abrupt end of this part.— 338 ways. 7). . 45. Brekhovskikh ELASTIC WAVES IN LAYERED MEDIA [13] pointed out that the effect of the velocity gradient due to hydrostatic pressure is quite analogous to the effect of the whispering galleries explained by Rayleigh (Chap. This part of the signal is the confined to the sound channel. For the duration T of a recorded signal at a distance r from an instantaneous source he gives the approximate formula (7-51) Da. and a is the ocean the sound velocity at the ocean surface. (a) Assumed velocity-depth relationship for ocean. -a o CO —I — 'z (a) \ \ < \. p. M(z.WAVE Haskell results [3G] PK01'A(.^)f^ Omitting the time factor and changing notation Eq. (7-20). general character of the assumed velocity-altitude relationship shown in Fig. 7-36 differs now from those. according to the first boundary condition. ^ of the source (r (7-o2j — > oo . (7-60) Taking two linearly independent solutions of Eq. consid(!red before. namely. the considerations which follow must be regarded only as an illustration of the method.ATION in MP:DIA WITIf VAHIAIJLE VELOCITY 339 hut his studied sound propafâtion in the atmosphere ^J'hus may be extended for the case of the ocean. we also put V=-Pi. at 2 ^= dz = (7-59).ssume that The (1) the vertical component of velocity vanishes at z = (the rigid bottom of the atmosphere). k) that behave asymptotically for large values of z {vrith the factor . k) and N(z. = h) the potential ip and (3) in the neighborhood must reduce to ^ ico = 0. ^ exp R' = r' -{- (z . and. Making use of the equation which holds approximately in this case. (7-55) we make use of ^ = Joikr)F{z) (7-56) This function must satisfy conditions (7-31): ^ dz -= dz = -2Ak F2 at 2.hf (7-53) a{h) being the local velocity. (7-18) again: slightly. We a. (2) the velocity of a particle V vanishes as z z = grad. atz = h (7-oi) and Fi factor F{z) is = = h (7-58) The in this case a solution of a more general equation than Eq. ag. k)M'{0. as was will Chap. except near z = ai. (7-54). /c„ k) [N(h.k)] Moreover. k) ^ mV^m) (7-65) = [ < Q dz (7-66) and C„ with is m = 2. (7-62) the solution of Eq. k) = shown in k. by integration with a If respect to the parameter be represented by sum of residues corresponding to k„ is the Bessel function Jo(kr) in (7-56) and some branch line integrals. ag z^. This a formal solution. . z^ be the values of z at co ^ which Q(z.{z.(2. Let c{k) 2 = co/k and J [c{k)' Qiz. O ||^ equals -M'{0. 4.k) [N(z. The solution < 2 < Oz. respectively. now is to choose the appropriate function N{z. k) = _ 2AkN{h. Substituting the expression (7-65) in Eq. k) . = -27rzAe'"' Z kM'o'\kj-)N{z. F = where . (7-59). k„) (7-63) dk The factor h now is N{0. (7-64) for c{k) -ik\l I L 'c{k)J Mz)j Z2. k) -1 1 for c(k) > a(z) = I L«(2). < a(z) Referring to Fig. b = MN' — M'N. which frequencies. a linear combination of Bessel functions of order \. in the ranges let ai. (7-62) leads to an estimation of some of the roots of the frequency equation. bN'iO. k) = < z < < z < is a solution of Eq.0. 7-36.k) k) M{h. replaced by the Hankel function. M' and are the derivatives with respect to Haskell showed that the boundary conditions are satisfied by F. For 1 is vahd in the range is m = the solution valid for < 2 ag except near 2 = aa. Has- kell's expression holds for the region above and below the level of the source. with m= 3 it is valid for z > a^.iz. the sum of residues ((pn) takes the form dN'iO. (7-62). are roots of the equation N'iO. k) = if 2AkN(z. k„) because of Eq. Haskell let Using an asymptotic approximation for high Z2 and < z < respectively. k) bN'(0.k)M'iO. - M(z. and the obvious question k). k)] (7-61) F. which is obtained. k)N'(. A'"' we now put z.340 e'"') ELASTIC WAVES IN LAYERED MEDIA like down and upward where traveling waves. k) ^.k)N'{0. WAVE PROPAGATION IN MEDIA WITH VARIABLE VELOCITY 341 placing the Using asymptotic approximation for the residues in Eq. (7-(')'4) and resums by equivalent integrals, Haskell evaluated these integrals of stationary phase. by the method The T Phase. A short period phase, T < I sec, traveling through the ocean with the velocity of sound in sea water, is frequently found on seismograms of island or coastal observatories for earthquakes in which the path of propagation is mostly oceanic. A typical occurrence is shown in Fig. 7-4. This phase was first noted by Linehan [40], and the mechanism ;Tmax. 60 sec. Fig. 7-4. phase from West Indian earthquake of Sept. 21, 1951, recorded by underwater seismograph at Bermuda. {Courtesy of M. Landisman.) T of propagation was established in the work of Tolstoy and Ewing [74] and Ewing, Press, and Worzel [265]. In the latter investigation, hydrophones in the SOFAR channel were used to detect earthquake-generated T phases. doubt that the energy crosses the deep ocean as sound SOFAR channel. However, compressional, shear, and possibly surface waves may be involved in propagation across the land segment of the paths. Recent work with Pacific T phases by Wadati and Inouye [76] and Byerly and Herrick [17] and with Atlantic T phases by Bath [8] supports these conclusions. Shurbet [68] reported observation of T phases recorded at Bermuda from South American shocks in which the energy was transmitted over as much as 51° as P, before entering the ocean at the scarp north of Puerto Rico. is little There waves, probably in the From and the that the known great frequency of occurrence of small earthquakes efficiency of propagation in the SOFAR channel it may be expected level"' in T waves contribute significantly to the acoustic "noise the and Sheehy [20] used T waves crossing the Pacific Ocean to study submarine volcanic eruptions off Japan. 7-3. Love Waves in Heterogeneous Isotropic Media. Since velocity gradients are known to exist in the earth's crust and mantle, it is very important to have a theory of Love-wave propagation in a medium where the velocity, rigidity, and density are functions of depth. The simplest case was investigated by Meissner [51], who considered a half space in which the density follows the law p = p'(l + 5^), where z is the depth, channel. Recently Dietz SOFAR 342 8 ELASTIC WAVES IN LAYERED MEDIA the case for 8zy. Wilson [79] studied a parameter, and the rigidity is ju = /x'(l = fi' exp (72;), jS = /3' exp (8z). Meissner [52] added a discussion + /j. fjL media in which one layer is homogeneous, the other heterogeneous. This case (heterogeneity in the lower layer) was investigated in more detail by Jeffreys [43]. He showed that for a mantle given by p = const, = n'(l 8zy, the heterogeneity increases the phase velocity and de- + creases the group velocity of the longer-period waves. A solution for a heterogeneous medium of such kind that the functions involved are of an elementary nature was given by Bateman [6]. One simple case of this problem was also considered by Aichi [4]. Sezawa and Kanai [66] found certain correspondence between the propagation of Love waves through some heterogeneous media and the problem of varying water depth, and Matuzawa [49] gave a solution of the problem on assuming linear changes in the z direction in both rigidity and in the velocity of propagation. Recently the case of a homogeneous layer over a substratum in which — hz, p = const, was discussed by Sato [63], and solutions for the H = in which variations in the substratum are given by ju = n' exp {8z), cases two "= p' exp (8z), and p. = p'(l -\- 8zy, p = p'(l -{- 8z), were given by Das p ijl' Gupta [19]. For Love waves, u = w = 0, v = v(x, z), forces the equation of motion for the y and in the absence of body component [compare (7-3r)] is dt dx dz By Eqs. (1-11) we obtain If M is a function of z only, Eq. (7-68) becomes If we put V = Vm V, this equation takes the form ^'^' 1 ^'^ V72T/ 1 r 1 (^"Y V (7-70) If we assume that ju = p(z) and V = the first Z(z)e' '"'-''' (7-71) factor will be the solution of the equation , 2 _ dz since /3 ^ . /^Y ^ '^ \lC\dzl , 2 I J_ L ^ 2p dz'J Z = (7-72) = (m/p)'- WAVE PROPAGATION Homogeneous Layer For /X2 = (7-72) in a form valid (Fig. 7-5). IN MEDIA WITlf VAKJAI5LE VEI/JCITY 343 over Heterogeneous m' Half Space: Matuzavxi's Case. - ^z, /^^ = /3' - 8z, Matuzawa [40] wrote Eq. for small values of z -p + dz where, after minor corrections, (A + Hz)Z = (7-73) 14' + 5^7^ - k' B 2 ^«' J + 4m' f7-74) and obtained the solution Z = where ^ r'.//;"(,|^r) + r.//s"(|^«*)_ (7-75) ^ = A + ^2 (7-76) z=H Ml Pi H ^ ^1 2=0 M2 = m'-&^ Fig. 7-5. Notations for space. Love waves in a homogeneous layer over a heterogeneous half (Matuzawa's case.) If a half space displaying such properties is overlain b}^ a homogeneous layer of the thickness H, the solution for this layer can be taken in the usual form (see Sec. 4-5): with 7i = \/c"/i8l — 1, and the l)oundary conditions for Love waves can easily be written. The period equation given by jMatuzawa for this case is tan /c/i7i = ^ 2 Hrky.A of the 1 \ 4-^1 ^ ^ Bfx'J + 1 fx' ^Z A H-2/3 ~ H4/3 2 MiA'Ti ttU) -"1/3 (7-78) where argument Hankel functions is 2.4 '/3S. If the asymptotic 344 ELASTIC WAVES IN LAYERED MEDIA expansions of these functions for large values of the argument are used, the approximate form of the period equation is tan kHyi = MS2 ixikji /5 2/ii/c7iS2 ^ h (7-79) 2iiikyi where si = —A. In Sato's [63] Sato's Case (Fig. 7-6). 2> problem the velocity of propa- k z=H: Ml Pi=P ^1 z=0 M2=1-5Mi P2^P ^2 z=-H2 H2 = n'-bz P3 = p >• Fig. 7-6. Notations for Love waves in a double homogeneous layer over a heterogeneous half space. (Sato's case.) gation in the substratum is 13^ = [(n' - hz)/pf, and Eq. (7-72) takes the form dz' + 2 "^83 k -j- Z = 4iU3 (7-80) By a change of variables y the equation = pco 2kh k = 2kiJ.a (7-81) d'W dr is 1 + + IW ^ ^ _ 1 ITF 4 = (7-82) obtained. It is satisfied by Whittaker's function wu^) = or e-^^rs = e-^^rii M^.oi^) + E (-1)" n ^^— ^- l^2 (7-83) --L-£/2 =^ê-''\FAh - y,h^) (7-84) - " WAVE PROPAGATION where ,Fi is the confluent IN MEDIA WITH VARIABLE VEI/JCITY 345 j hypergeometric function. By Eqs. (7-71 and (7-83) and V = ., y/n^Vi we have $(^) = a(A)^*(^).-' = f -ie-^/^s (7-85) elastic properties just defined, Considering two homogeneous layers overlying the half space having Sato makes use of the expression (7-77j for each layer. If quantities referred to 1, 2, the layers and half space are denoted the boundary conditions by the subscripts be written as and 3, respectively, may ^= dz If atz = H^ = Vz V, =v^ and î ~= ~ oz M2 at 2 = (7-86) az ' V2 and 1, M2 = Tdz Ms at 2; dz is = —Ho (7-87) we put y] = c^/î the period equation obtained in the form 1 - ^ M272 tan kH,y, tan kH^y 2$'(U 724>(y + <^ tan kH,y, + ^^ tan kH,y, Im (7-88) M72 where Ms = m' — ^(2! + ^2) and ^0 = 2kfji' (7-89) The phase and group Hi — H2 = velocities were determined by Sato for the conditions H M2 = 1-5/xi = 1.5 Pi "' = P2 "-' = p " hH = 40 His phase- and group-velocity curves are presented in Fig. 7-7 Phase veiociiy i 1.0 ^ 0.8 ^ ^y^ ^ ._ -- — ^ Group velocity ^^ "^ ^^ <^C— ^^ ^ y 0.6 Period, unit:H/-vAi'/p Fig. 7-7. Phase- and group-velocity curves for Sat6's case. 346 Jeffreys' ELASTIC WAVES IN LAYERED MEDIA Case (Fig. 7-8) . Jeffreys [43] considered the system of a homogeneous layer of density pi, having a uniform density p2 q being constants. If we put composed and thickness H, and the half-space and the rigidity ^2 = m'(1 + ^/qY, m' and ?= and y 1 + = a2 <- 1 2 72 for 1 < ^ 2 (7-90) = (A cos ^-pi^ + B sin A'7i2)e '*"""' V -^ e < (7-91) Ve'''^-'"' for < (7-92) Equation (7-68) for the second medium k^q[l takes the form ^{m z=-H ^1 - ^^.j^V = (7-93) 2=0 P2 M2 = jU'(l+^)^ 2Y Fig. 7-8. Notations for space. (Jeffreys' case.) Love waves in a homogeneous layer over a heterogeneous half Two cases have to be considered according as c ^ ^2, the boundary conditions being of the type used before. The final (32 form of the period equation obtained by Jeffreys tan kHyi M2T2 MiTi L is as follows: (1) c < [see also Eq. (B-22)]; 2kqy2 \ 72 where ^2 ^-^Kyl 2 (1 0, 'Y2)(5 + is, 72) + Oikqy'j 1. (7-94) = n' is evaluated at = that ^ = Since asymptotic solutions of Eq. (7-93) are used, the approximation depends upon the largeness of ^572. (2) c > 02- If we use the notations cosh 6 sinh = f^ = (|J - 1)' (7-95) WAVE PROPAGATION IN MEDIA WITH the period equation is VAKIAfiLE VELOCITY 347 (for ha-go values of kqyi) tan kH'yi Mi7i 72 cot \kq{d cosh 6 — sinh ^) A~ ~ Vi \ j 1 - ^V i (7-90) 2kqyl J Since moderate values hjy'l or kqyl are also important, JflTreys derived a third form of the period equation. Using the new variables ^' and ^" given by the equations ? = ^(i +r) (7-97) (7-98) he found that Eq. (7-96) becomes (7-99) where the Airy integral. Ai(x) =TT Jo f COS kx ~k']dk (7-100) is Calculations based on Jeffreys' theory have led to dispersion curves which have been compared with experimental observations of Love waves (see Fig. 4-52). In the problem investigated by Jeffreys the densities are assumed to be constant. However, variations of densities must also be considered in some applications. Meissner's Case (Fig. 7-9). Meissner [51] considered a half space in 2=0 p = p'il + dz) l + tz 1+52 i3î8„, as 2 zi Fig. 7-9. Notations for Love waves in a heterogeneous half space. (Meissner's case.) 348 ELASTIC WAVES IN LAYERED MEDIA which p = p'(l + 52) M = m'(1 + 62) (7-101) and, therefore, The hmit value of the velocity at infinity has the finite value 13^ = lim m= /3\/-^ (7-103) If we put V = 2y-{l +ez) y = Jl - ^-, o) = kc (7-104) Eq. (7-72) takes the form where p = |J| if (l - ^)(l f^'' (7-106) Now, we put is 5 = 0, we obtain the case considered by Sato, and Eq. (7-105) of the type (7-82). We can consider the solution of Eq. (7-105) in terms of Whittaker's functions, Z = AW,,Uv) and find the exact + is BW.,,,,o(-r]) (7-107) form of the period equation (indicated by Jardetzky). The boundary condition for this case p^^ = ^ = at z = (7-108) Since surface waves are involved, we have Z{ri) as 2 ^ The oo lim 7, = 0. (7-109) solution which tends -.CO This to last condition requires that as 2 -^ 00 is, then, we put B = Z = AW,y,M where (7-110) . A is a constant. Now V = V/\/n, and z, by Eq. (7-71), is if we omit factors which do not depend on the condition (7-108) 4-5. we have from (7-100) Finally..„. of the WTiit- For the case of a half space in which p = p'(l + dz) fx = m'(1 + 5z)' (7-118) Meissner obtained a solution by means of the Laplace transformation.WAVE PROPAGATION or IN MEDIA WITH VARIABLE VELOCITY 349 2m^|-Z^ = f7-112j By Eqs. eliminate the two variables p and 7/0 from Eqs. (7-113) = 2(1 — c^/^^y k/e.oiv) -f 07] - W.Jl - '-. Observations of Love-wave dispersion from earthquake seismograms were discussed in Sec. (7-104) and (7.o(v) = at 2 = 0.2.n. 7. On the other hand.ilO) we express the required condition (7-1 I2j as the period equation in the form 2ri W. Das Gupta [19] solved this problem for a homogeneous layer underlain by a half space of the type (7-118). that determined by in surface-wave studies. where it was shown that inclusion of the effect of the velocity gradient in the structure determined mantle was necessary to reconcile cnistal by explosion seismology \xith. one can use the asymptotic expansion taker function. Rayleigh Waves Heterogeneous Isotropic Media. cussion of Rayleigh-wave propagation in a heterogeneous The first dismedium seems . using Whittaker functions.)] dt = (7-117) As an approximation. (7-113) by c is using (7-114) and (7-115) to obtain an equation in which in terms of expressed Since Whittaker's function Wy. if I = 27r//c is the wavelength. then d = . 7-4. = 27 € (7-113j To relate phase velocity to relate the variables p and 170 wavelength we first note that Eqs. is given by the formula the exact form of the period equation in this case e-H-''''''\t is I + 77o)"'~'^''[p - 1 - (^ + V. (7-115) We can now I. 350 to ELASTIC WAVES IN LAYERED MEDIA [40]. for /x = cs the solution could be "expressed in terms of the confluent hypergeometric function. where z is the depth. all members of Eqs. Considering the case ii = ixq -\equations of motion in the form -^ + T+ mV w + Mi^ \dz dx/ dx 611 --- + + I pkcu = (7-123) 7 dz -f mV w + 2 . Sezawa [65] investigated general queswave propagation in a semi-infinite solid body of varying elasticity. o 2^1 ^^ -oz 2 2 pfc c w = n .. N cos\ >mx-e smj . reduces to well-known types. X -^ folloAving CX3 >» = ^ + |^ dx az jjlx (7-122) Love (Chap. (7^) were expressed in terms of cylindrical components of rotation co^. Suppose that xi by the factor exp \ik{ct — x)\ and put u^'l-'-âz OX Now. Stoneley [72] took the equations for two-dimensional motion which represent a particular case of Eqs. 26). With external forces omitted. (7-4) (7-121) and and also w are functions of z multiplied assumed that the medium is incompressible.Three equations for these functions could be satisfied by expressions of the form have been given by Honda tions of w.z). clz ^ $' + (^ _ A^ \ ji = (7-120) I This equation. o:. iî where #(2) is the integral of the differential equation $- + 1 y. 3. Ref. taking 6 = 0. assume that —11 = lim \d as z. w^. {kr) NT7'2)/. He assumed incompressibility. and m = c (d -\. the component w equal to zero. in cases of the simplest laws for rigidity distribution. Thus. we now have the and 0-^0. = B„îz)Hl:\kr) ^^^\mx-e'"' — cosj iOy ^ = B^7n ^-^rkrr ^ (z)H„. who also extended the investigation to compressible media.WAVE and If propa(. For a heterogeneous layer Newlands solved the second directly instead of using Whittaker's fimction.i5] = 10 -\- pkV<p (7-124) we put F = Eqs. For a surface wave. Rayleigh waves in a semi-infinite incompressible medium where a layer in which rigidity varies linearly with depth is underlain by a uniform elastic substratum were investigated by Newlands [55]. He also considered the problem of a uniform surface layer overlying a heterogeneous half space. (7-128) a if we put = 1 + this equation takes the form .„/p)K k/xo/fJii. only the negative exponent in the solution of Eqs. asymptotic expansions for these Whittaker functions cannot be applied to this case. and Stoneley investigated by other methods the two limiting cases valid for short or long waves. and the two functions <p and i/- are taken in the form = Ae-'' cos k{ct - x) ^ = BW^. (7-128) to the form $H-(MV = which is (7-129.ati(jn in II media with 2jLti vAJiiAiiij-: veujcity . satisfied by Whittaker's function (p Wa. where h reduce the second of Eqs. (7-128) will be used. fjiiz/no. we have == for the velocity at the surface ^q The then substitutions ^ 2{kz + = hc^ /2^l. 1 ^^' ]-'( iV = = = (7-128) the interior {iJ. Thus. Since the free surface of the is 2 = 6) and the and a z axis is directed into body. (7-123) take the form ijlV'4^ + pkVrP = (7-12oj dx dz (7-1 20) and if we write <P = 1 1 ik{ct—x) <P()e (7-127) we obtain (on omitting the subscript 0) d-<p 2 d^4^ f. of Eqs.î^kz + 25) sin k(ct - x) (7-130) Then the boundary conditions yield the period eciuation 2 - l](^- + 2 - ^-l - ^ - ^^® (7-131) Unfortunately.i(^). namely.xl^''' (7-136) where '^"' = ^ + (^)''^^'' + • '^'" ' + (tt)''^^'' + • • • (^-1^^) By the second equation in (7-134) we get U^) = and it if f d^ f (1 . with logarithmic Now. if two layers are considered.4^^'\d) -\- Q sinh kz) cos k(ct — x) . (the axis taken positive downward) and vanishing of stresses at the free surface z = —H yield six conditions for the constants A^. R. and S.' = {l . lA. ••• . is A 2. K.^)'A„ (7-134) By (7-134) one can take ô(S) = A.__ „„. = (1 . (7-132) ^" = r. The continuity of displacements and stresses at the interface 2 = P.| P2 ^.. + A 2xl^''\d)] sin kict re) x) and for the homogeneous substratum (P2 = Re'"' cos k(ct (7-140) x) ^2 = = »Se~'^^'^sin k{ct — = where ^^ i .^)u<r) da (7-138) we may use the absolute value of this expression as well as the other \l/„. (7-133) = Ud) + (77)^1(5) + • • • + (^)'V«(S) + • • • we obtain from Eq. be proved that the series converges and that ^^^\?>) and i/''^'(^) d can be written in terms of certain polynomials in factors. Q.352 ELASTIC WAVES IN LAYEEED MEDIA integral in a series of Expanding the 'A(S) powers of {k mo/mi)^. ^P = AiV'''' + = A. + A^z (7-135) Then.^^0. ^ P2 (7 141) and P2 and is jLt2 are the density and the rigidity of the substratum. Since this system a homogeneous one. its determinant must . from the linear differential equation (7-132). we can assume that for the upper layer the solutions of Eqs. (7-128) hold: <Pi = = (P cosh kz [A. dx if When the additional constants are neglected. 7-10) showing the long waves were . These above. as before. ~= dx (7-143) /2 = mV-iA + + 2mi ^ = where the factor of the (p mi is d^/dz and the last terms are due to heterogeneity medium. makes use of Eqs.WAVE PROPAGATION vanish. Trains of long-period Rayleigh waves which have circled the earth several times are often observed on seismograms this transformation crust where both n of the greatest earthquakes. one expands the functions and ^ into series <p ^ ^ ^(3) = Fo rP ^P{^) Go ^ +^ + Ml Ml Fi + + kjxo ^2 Ml + (7-144) (?: (?2 + Ml where the variable 3 is 1 miVmo. and Newlands suggested that an adjustment of the assumed constants was needed. (7-143) for successive computation of functions i^. the dispersion of these waves has been studied in sufficient detail to extend the Rayleigh-wave dispersion curve to the range 70 to 480 sec. G. With the aid of modern long-period seismographs. (7-121 j. If the medium is equations take a simpler form for exp (rb^coO. Following Newlands' method. X also becomes a function of coordinates. IN MEDIA WITH VARIABLE VELOCITY We thus have the period ecjuation for the medium determined compressible.. Remarkable seismograms (Fig. namely. (7-142) are satisfied /i = (X + 2m) W+ p/cViA pkV + pk c (f — 2/x. dyl/ — dx + d_ dz mV'iA + pk'^cyp + dtp 2/i. Comparison of the computed group-velocity dispersion curve for Rayleigh-wave obser^•ations of Rohrbach [59] did not show any agreement. Mantle Rayleigh Waves. we have to make use of Eqs. Eqs. = (7-142) dx (X dz + 2/x)VV + pk^c^<p — 2ni dx d_ dx juVV + pfcViA + dip 2/i. The results of + were applied to the problem of Rayleigh waves in a and X vary linearly with depth. and. and. (^ 354 . linear-strain Rayleigh-wave trains of orders for Rr. proper for Rayleigh waves. The short-period limit T 75 sec corresponds to the least wavelength for which the continent-ocean margm is a neghgible = barrier. where n is the order and A the least distance between station and epicenter. and the flattening of the curve for periods greater than 400 sec." As a rough approximation. the constants being chosen to fit the observed km/sec.WAVE PROPAGATION obtained from the IN MEDIA WITH VARIALSLE VELOCITY 3o5 Kamchatka earthquake of Nov. hence the use of the name "mantle Rayleigh waves. The dispersion of crustal Rayleigh (velocity increasing with period) . Ref. For periods greater than 75 sec. there can be no doubt that the dispersion is the result of the known increase of shear velocity with depth in the mantle. 4. using the methods of the preceding section. to /?. Period and arrival time were read from the record in the usual manner (Ewing and Press [27]).8 km/sec at 70 sec. the mantle velocity gradient may be replaced by two homogeneous layers and the theory of Sec.15 km /see. Since the two dispersion curves must merge. Observed group velocity for the long Rayleigh waves is plotted as a function of period in Fig. 7-1 1 for (in corresponding to epicentral distance = {n — n odd. Comparison of the strain and pendulum seismograms from Pasadena demonstrated that the orbital motion was retrograde elliptical. 4-5 used to compute the theoretical curve shown in Fig.52.54 minimum value of group km/sec at a period of 225 sec. A theoretical curve (Haskell. but these calculations are very lengthy. using the epicentral distance appropriate for the number of circuits of the earth. In view of the great length of these waves. 7-11. waves for T < 50 sec is normal and that of mantle Rayleigh waves for 75 sec < T < 225 sec is reverse. Chap.A n even. 7-12. a short-period limit of 3. a maximum value of group velocity must exist between periods 50 and 75 sec. A complete Rayleigh-wave dispersion curve including that for oceanic and continental crustal Rayleigh waves and also that for the mantle is sho^vn in Fig. 2. on the lieniofT seismograph at Pasadena and on the long-period verticalpendulum seismograph at Palisades (Ewing and Press. and from these data the group velocity was calculated. and A„ = n • 1 80 — A for the various orders. and minimum group velocity.48 = 516 km.. /3i = 4. degrees) A„ IjlSO 4. Ref. Chap. oceanic and continental paths cannot be distinguished.-. 19. are indicated on the seismograrn. Striking features are the velocity of 3. 12). Several important results emerge from the study of mantle Rayleigh waves: H 1. 4. 62) was used with the constants ^2 = 6. 5. Better agreement with observation could be obtained by computing theoretical curves for a two-lajêr heterogeneous medium. Group. 50 m R5 4. = 215 (sin tude factor 8 = tt/QcT.6 3.48 km.6 4 them from the effects and ampHtude measurements can be made with 15 2 Mar. = 516 km. for mantle Rayleigh waves obtained from Pasadena sufficient precision for studying decrement.4 100 200 Period in 300 seconds 400 500 Fig.15 km. 52 Aug. 4. 33 Sept. it was found that 1/Q = 665 X allowance for geometric spreading by an ampHA„)"' and dispersion by a factor A~^ The amplitude A where sec. after . For an amplitude decrement given by exp 10 at 7" ( — 8 An). /sec.8 3. 7-11.0 Theoretical 02= 6.356 3. /sec. (Haskell) / 5' 3.velocity curve and Palisades seismograms. great length of mantle Rayleigh waves frees of crustal irregularities. 4. /3i H'i = 4.4 R2 R3 O PAS o PAS bPAS X DPAS + PAS PAS • PAS PAS 48 PAS - R6 R7 R8 R9 PAL PAL PAL PAL PAL A PAL ^ A PAS mo = ffn ^ R]? 4. ELASTIC WAVES IN LAYEKED MEDIA The The flattening of the dispersion curve for T > 400 sec is interpreted as an effect of the vanishing rigidity of the earth's core. Nov.2 *PAL R13 R14 R15 liPAL « PAL ^ PAL \ 4. i E 10 o o p k 1 Press 19 iwing— 'ress ^x and Press. — 1 and Ewing [wing al Continent Mantle— Oceanic- 0< ° * > . o o * 5 CO => o rv j5 / S o c c o "c OJ O o o « e e I e 1 1 1 1 1 1 1 ' 1 1 1 1 1 1 1 r 1 r 1 1 o •09S/-UJVJ 'AipoiaA dnojo fo 357 . p. 34) in the preceding sections. for example. the so-called Fresnel law determines the velocity in any given direction in terms of these three velocities. 2Nê. p..- CiiBii. iVl^xz Pzx (7-148) u and w expressed in terms of Then he could show the existence of waves of the Rayleigh type. are considered as hnear functions of strain components Cj:^. exponential functions.. who published his results in Japanese in 1942. Rq being the earth's radius. Chap. the first attempt to investigate wave propagamedia seems to have been made by Homma. three principal velocities of in wave propagation an aeolotropic medium. • • • . ilfi(e„ + e„J + ej 2iV0(exx = (M3 - + (M3 2N.. p. If. •'• i. It is well known (see. the assumption that the media are isotropic is a reasonable one. and ^ = for other periods in the ndf ^-i*« wave train. he obtained the period equation. For elastic waves in aeolotropic media. as usual._e. Pxx = 2Z c. its boundary being a plane surface. 2Aî)e„ Py. e„. Aq and In the investigations discussed Aq constants. isotropic elastic that there are.^ 358 is ELASTIC WAVES IN LAYEEED MEDIA given by ^ = for the IrXIu^ <^-"5) Airy phase. y. for example. on taking the usual form of solutions. He assumed that 2A^0e. Sato. the stress components p^^. = = Nê^. 7-5.... in general. Some attempts have been made to explain some discrepancies between observation and theory by assuming aeolotropy in rocks (White and Sengbush [78]). media were assumed.j = X. p... Matuzawa [50] made use of results of Sakadi [61] concerning wave propagation in crystalhne and. coefficients reduce to a smaller number for different types of According to Sato [62]. Love. For electromagnetic waves. + (M3 + ej + M.. = = ilf i(e. we can write • • • . discussed a two-dimensional problem of wave propagation in a medium which is tion in such horizontally isotropic and vertically aeolotropic.e. making use of these results. Aeolotropic and Other Media. Ref. 1. Since most igneous rocks are crystalline and the crystals commonly have random orientation. . or z (7-147) The 36 crystals. one has to make further assumptions about the nature of the medium. A theory of wave propagation in the so-called orthotropic media was [18]. + . Rayleigh and Love waves can be propagated over the surface of a "transversely isotropic" body. given by Carrier are in Since there are layers of porous material through which seismic waves many cases propagated. + L{C + eL) + Nel. There is no sharp distinction between the dilatational and distortional waves if a disturbance is propagated in an aeolotropic medium. to account for the existence of first empty holes or holes filled with liquid in a solid medium. and solutions first. when the holes are . (7-151) if .— WAVE PROPAGATION media. . N are constant and . Ref. An explosion in such a medium will produce both P and 8 waves. This body or medium is determined by the condition (Love. and p that of an actual material.---) dt dx dz dx dz (7-152) d^W P -T-. as was pointed out by Stoneley [73]..Je„ + 2(A . the coefficients A.ê.. IN MEDIA WITH VARIAHLE VELOCITY 3o9 The stresses crystals.2N)e. 1.. L) — dx d\ dz Investigations of media with other physical properties have been published recently. the eciuations of motion (1-7) take the form (p^^ = dW/de^^. + e. {F .) If we assume that + CeL + 2F(e. dt d^W ^ d^W —-2 -^ C -—2 dx dz . p. is similar to a dilatational wave for the first root of the velocity equation and is more a distortional wave for the second root. The were taken in the form corresponding to hexagonal of equations of motion were given for two kinds of determined by the conditions -\- u = J{my nz — wl) v = w = The second (7-1 V. po being the density of the material without holes. i.. body forces are omitted. We shall mention here only some of the definitions of such media and the results concerning the propagation of disturbances in them. attempts were made [32]. 34. for which we assume w that y = /i = f\{my -\- nz — wt) w = fîmy -\- nz — like wt) (7-150) where and /a are arbitrary functions. kind. C. Chap. waves.2 = L . The case was considered by Frohlich and Sack and Mackenzie 5 [47] expressed the elastic constants in terms of the relative density = p/po. 160) that the strain-energy function has the form 2TF = A(eL + el. on assuming that the holes are spherical.„ + .e. as was shown by Stoneley.)) displays the character of a distortional wave. 343-348. Bateman. Ann. Brekhovskikh. Bull. Nauk S.: The Propagation of Sound in an Inhomogeneous and Moving Medium.. Am. vol. L. 28.S. 62. in a Medium with Variable Index of RefracAmer. /. 137-142. 4. vol. vol. Blokhintzev. 18. M.S. 1954. 1950.S. Elec. vol. 9. Japan. Anderson. vol. 93.360 ELASTIC WAVES IN LAYERED MEDLA On making use of these results. Brekhovskikh. Aichi. 469-471. 3.S.: The Elastic Waves Lg and 295-342. pp. Abeles. 17.^ . F. 1950. L.S. Math. 5. tion.. U. G. III. Doklady Akad. pp.S. 1949 (in Russian). 2. Physik. R. with Applications to Thin Films. vol.: Distribution of Sound in an Underwater Sound Channel. pp. vol.. Nauk S. Soc. 596. Acoust. Nauk S. Byerly. II. REFERENCES 1.. pp..R. Math. On the Transversal Seismic Waves Travelling upon the Surface of Heterogeneous Material. M. Rg along Euroasiatic Paths. : 279. Geophys. 2. Doklady Akad. 221-228. Sato [64] computed the velocities and S waves in terms of the porosity (1 — 5) for empty holes as well as for holes filled with liquid. 1931. les Phijsik. Tech. 15. vol. The Propagation of 160-165. M. Herrick: T Phases from Hawaiian Earthquakes. vol. Akad. 17. Akad.S. 1948..: Distribution of Sound in a Liquid Layer ^-ith a Constant Velocity Gradient. p. pp. Nauk S.: Distribution of Sound Velocity in a Section of the Eastern North Pacific. : Waves in Orthotropic Media. 1949 (in in a Russian). /.. of P those for the corresponding media. J. 1946. 6. pp. vol. Izvest. 14.R. 505-545. vol. G. 12.S.: The Reflection of Plane Waves from Layered Inhomogeneous Media.: Elements of Radio Meteorology: How Weather and Climate Cause Unorthodox Radar Vision beyond the Geometrical Horizon. vol. M. pp. Wave propagation in granular media is now the subject of numerous experimental as well as theoretical studies. (London). vol. Izvest. B&th.. Abeles. H. E.: On a New Method to Determine the Field of a Point Source Layered Inhomogeneous Medium. 8. 13. Amer. vol. pp. 1946. Research on the Propagation of Sinusoidal Waves in Stratified Media. Soc. F.S. The Propagation : of Electromagnetic Waves in Stratified Media. 5. K. 4. pp. Booker. Brekhovskikh. Sur la propagation normale des ondes electromagnetiques dans opt. pp. P. 1126-1135.: The Wave Equation 13. F. Seism. 69-78. 34. G.: Sound Rays as Extremals. : 1928. pp.: On the Field of a Point Source in a Stratified Inhomogeneous Medium: I. 13. pp. vol. Soc. 1946. 157-160. H. L. M. and C. Bergmann. Am. Bateman.R. Amer. Amer. 2.. 10. Ann. Carrier. 329-333. L. Inst. 3. 1948 (in Russian). pp. 18. vol. 1954. pp. 19. 44. 113-122. H. vol. 1922. milieux stratifies quelconques. J. Appl. P. 1949 (in Russian). I. 1949. Soc. Quart.: Abeles. 1946. IIIA. p. Brekhovskikh. Math. Arkiv Geofysik. Brekhovskikh. Rev. 504. 11.R. Proc.S.. Engrs.R. vol. 69. 7. 16. pp. Acoust... Acoust. L. Phys. M. /. 1949 (in Russian). 322-334. 4. Trans. 468-475. 409-420.: Transverse Seismic Waves on the Surface of a Semi-infinite Solid Composed of Heterogeneous Material. 31. p. Soc. Union. Bull. These velocities have smaller values than empty. Soc. F. 1951.50. F. Geol. Acad. 32. 1-10. 57.. 1. Seism. Physik [5]. Homma... 1952. Press. Press: An Investigation of Mantle Rayleigh Waves. Phys. Soc. 121-124. 22. 1953. 1951. M. K. Vine. W. Fortpflanzung des Lichtes durch ein inhomogenes Medium. pp. Frohlich. vol.: On the Wave-guiding Properties of Heterogeneous Media.. vol. pp. 127-148. Woollard. Soc.. 1948.. 265. vol. Physik 709-736. 33. 27. 1930. P. Swainaon: The Velocity and Ray Paths of Sound Waves in Deep Sea Water. 16. vol. 225. Ann. S. Geophysics.. Haskell. Bull. Dietz. Soc.A. pp. Amer. M. R. Amer. 8. : 40.. Geofis. 29. and F. 29. pp. and O. vol. vol. G. Inst. Geol. 1951. E. Natl. Bull. 34. 1915. 23. Tolstoy. pp. 627-637.. M. pp. 28. Mag. vol.: Uber die Ausbreitung des Lichtes in inhomogenen Medien. 18. R. pp. 237. On the Propagation of Dispersive Seismic Waves. 75-103. A. C. vol. Amer. Seism. U.ro- Das Gupta. A. 24. Ewing.S. 37-51. Geophysics. 209-236. 185. L. 15. 1930. Epstein.. 37. S. K. B. (London). U. and R. vol. sions Proc. J. Press: Proposed Use of the J. Ewing. 39. p. Ann.. Geol. D. 25-54. B. 1954. Krasnushkin.. 21. 1'U0PA(. Mining Mel. S. (Tokyo). vol. 909-934. Phys.. Dyk. pp. Zi. 65. Engrs. Seism.: Kisyo-syiisi [9].: Effects of Low-velocity Layers. pp. Soc. M. 1913.: Reflection of Waves in an Inhomogeneous Absorbing . pp. Physik. Solomon Islands Earthquake of 29 July 1950.WAVK 19. 1946. 35. J. Asymptotic Approximation for the Normal Modes in Sound Channel : Propagation.S. Appl. geneouH S. 431-446. J. 44. Seism. pp. Ewing. and M. vol. N. A. pp. Mem.. Ewing. H. Tech. 127f)-1277. P. 283- 36. 30. I. Ewing. and Worzel: Further Study of the Phase. 1952.. pp.ATI0N in media with variable VELfJCITY a IIomogeneouH CruHt Laid upon 3G1 lii-X'-. 1953. 1950. Forsterling. II Irulian J. vol. vol. 1-39. and F. p. S. Forsterling. 1954. Am. 23. 3-18. Phys. pp. 42. 1931 (in Japanese).: Channel Waves in the Earth's Crust.Medium. Bull. 11. 1952. vol. and J. 31. vol. 1914. vol. Tech. 18. C: Note on Love Waves in Medium I. 266.R. and L.. 20. 129-133. [4]. and 25. 1954. Appl. Rheehy: TranHpaoifio Detection of Myojin Volcanic Exploby Underwater Sound. pp. 315-325. and H. P. 1931. Physik. Amer. inhomogenen Medien: I. pp. Press: Crustal Structure and Surface Wave Dispersion. 19. M. vol. Bull. (Tokyo). 22. Leet: Seismic Propagation Paths. Geophys. 40. Honda. II: Soc. 47. pp. 1953. L. 157-168. Geophys. Phys. Roy.. Sack: Theory of the Rheological Properties of Dispersions. pura e appl. .. 53-58. pp. 27. N. M. Air Force Cambridge Research Center Geophys. Z. 294.. vol. 42. 1955. Forsterling.S. 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The Seismological Implications of Aeolotropy in Continental Structure. vol. Kanai: On the Propa^âtion of Raylf. 1953. Kanai: On Shallow Water Waves TranHrnitter] in tho iJireotion Parallel to a Sea Coa.. and M. 1955. 81. pp. schweren Korpern: I. Inst. pp. vol. pp. o^'. 29. D. Seism. Gerlands Beitr. 76. 15. Sci. 77. 42. vol. Bull.0-694. pp. Sengbush: Velocity Measurements ' : in Near-surface Forma- tions. 5. Hoc. pp. Bull. J.. with SptK-ial R(!f(!rence to Love Wavf.: Sci. 78. SHsm.: 73. pp. 80. Acad. Bull. In. R. 19. Pvhl.30. Waves in a Heterogeneous Medium. Bull. K.R.WAVE PROPAGATION 66. Japan Acad.. vol. 1941. and O. T. pp. K. Yosiyama.tf. Amer. 6. Monthly Notices Roy. 79.: Sur I'equation d'onde pour 2. K. 41-53. 1934. 1941.. Sobolev. S. 19. 11. le cas d'un milieu heterogene isotrorx. 3. Soc. 11.V. Arner. 11.S. II.. Wilson.u heterogene.i-o'j'. Amer. Wilson.S.. . of the formal integral solutions obtained in Chaps. because of the nonperiodic factor is exp (xp). for determining Eckart Jeffreys and Jeffreys [4]. the largest values of the integrand occur where p of steepest descent is large. both p and a are functions we put f = fc We can visualize the connection between these + variables analytic. Nakamura (Chap. remainder of the path may be neglected in an it. due to Debye. Of interest paper by Honda is a Sommerfeld (Chap. Since /(f) is dp # _ 'dk~ Yt _ " ~dk ^ where f^ d7l C^-'^') An extremum or a saddle point occurs at points ^ dk = |£ dr = or ^ ds = = (A^) 365 . critical distances The method of steepest descent is applied to the evaluation of integrals of the general form I{x) = j F{^)e^'''' rff . and 1. The basic idea of the if method to deform the path of integration. According to the latter. the name "method of saddle points" or "pass method" is more appropriate.APPENDIX A METHOD OF STEEPEST DESCENT This method. 56. in order that the approximate evaluation. by representing p as an elevation over the da §_P J~T f plane. 68) and 4. by the in which solutions obtained Ref. the method see discussions of For [2]. particularly certain where types of waves first appear. positive. F(^) compared with the exponential factor. possible. and /(f) is an analytic function. p. 99). in such a way as to concentrate the large values of p in the shortest possible interval. and 4. method of steepest descent and by use of the Sommerfeld contour are compared. (A-1) and real. of the variables k If and r. varies slowly imaginary parts /(f) = p + ia (A-2) and note that. Ref. and the integration follows a path in the complex f plane. may be used to advantage in the evaluation 2. We separate /(f) into real and where x is large. 3. r) = const vanish. without altering the value of the integral.There are at least two curves p(k.9p ^ _da ds r) (A-^) dn = const represents a family of curves. t) ^ ^ dn . However. r) = const. since V^p = 0. At any point of the curve p(k. r) = const. = const. and in this case the four sectors between the curves are alternately hills and valleys of the surface p = p(k. t) = const represents a line of steepest descent. In order to concentrate large values of p in short intervals of integration and keep p as small as possible elsewhere. Near fo. along which the largest changes in p occur. (A-4). r). where — f forms with the k axis. the maximum variation of p occurs along the normal. one such line occurring in each valley and terminating either at infinity or at a singular point. Thus a curve of the family <j{k. Suppose now that it is possible to modif}^ the path of integration in (A-1). in such a way as to pass from one valley to another through a saddle point fo and coincide with a(k. the path and keep to the valleys. o For we have. along a path of steepest descent where \dp/ds\ is as great as possible. it must do so through a saddle point.366 that is. Put fo)V"(ro) (A-7) W~ = and change the variable of integration in (A-1) to k to obtain approximately (A-8) r is 7 = e-^'f»' J f FU)e-"^''^ write T UK ^ cIk To find the factor d^/dK we can small and real and x is the angle which f f on the path in the neighborhood of fo ~ f o = ^ exp (^x).f(f) = /(fo) is + Kf -Kf - ro)7"(fo) + • • • (A 6) where the second term negative and real along the path. only saddle points are defined by Eqs. ^ve obtain K = ±r |/"(ro)|^ = ±(f - ô)e-'' |/"(ro)|^ (A-9) and 1= ±e-'Mr(fo)|^ (A-10) . Since by Eqs. (A-7).It is obvious that the neighborhood of fo yields the largest contribution to the integral. at least in the vicinity of f o. Since k^ is real and positive. /(f) can be expanded in the form . k^ = — r^f'iô) must be dr/d^ exp (2ix). Since arg ( [/"(fo) exp (2ix)] must be ±7r and = exp — z'x). r) = const. (A-3) we have must avoid the hills §£ ds a(k. ELASTIC WAVES IN LAYERED MEDIA where the derivatives in the tangential (s) and normal (n) directions to the curve p(k. If the contour crosses from one valley to another. Saddle points are also stationary points of a and zeros of /'(f). orthogonal to p{k. by Eq. the coefficient of — r^ real and negative. we can write as before I F{^)e^''' dr = F{U)e'''''' f e^^'" ''''''-'''' d^ (A-14) where the path is chosen such that the exponent in the integrand is purely imaginary and the limits are extended to ±03. the limits oo and use can be made of Watson's of K may be extended to the range ± . At places where not vanish and is are nearly /'(f) does imaginary.„ that value of x is selected which then the plus sign in Eq. If f. the factor exp [.MIOTHOD OF STEEPEST DESCENT 307 tt) To To select the proper value of x. Jeffreys and 473]) by writing I (A-8) in the form ! e-^'"'F{-^)e'' dK (A-li. p.symptotic expansion of (A-8) can Jeffreys [4. integrals of the type (A-1) by using paths through saddle points such that p. The integral on the righthand side then reduces to a form which can be evaluated./4 according as f" ^ (A-lo) Although a mathematical proof of the method of stationary phase available (Watson [7]). = <T{k. Thus ^^'^^^ !x/"(ro)|^ If it is of integration of (A-8).. In Kelvin's method of stationary phase. it is usually justified in terms of interferences is of wave motion. phases the same. (A-10) makes is r positive after passing through taken. is constant. (see. now be obtained E(]. for example.rf (f)(f — To)] in the inte- . The first term in the a. Since /'(D = for both cases. In the vicinity of stationary values of /(r).„) ( — tt.„ t„). For this path the modulus of exp [xf{^)] is constant are evaluated while the phase varies. giving VgrFJroK^ *i. The is condition = is used to determine = K + ^V. necessary to pass through two or more saddle points in the path then each saddle point gives a contribution to the integral. then tan x the slope of the path a{k. the more closely the higher values of the integrand concentrate about the saddle point. lemma f e-^'-'M where <po dK ~ V2^ (^ + • • •) (A-12) is the first term of the ^ series expansion of <p{k). and the contributions are additive. For large values of x. = l/"(fo)| The larger the value of x. i'l the raiifê f'{i. rather that a.. two values t) of x differing by tt are possible. is grand and the resultant contribution to the integral small. (A-20). 4. provided that contributions from any singularities crossed are taken into account.368 ELASTIC WAVES IN LAYERED MEDIA oscillates rapidly for large values of x. namely. Ref. saddle point at fo. this constant a To determined by Eq. 68). we have = ukd. .k%f of the and m and The n are linear functions of is z. that is. where It can be proved.Y^ = {f . 2. = s - ic == {f - kl. /(^) = v. where three valleys meet at the saddle point. lies on a path methods discussed by Ott [6] are necessary. for example. and Honda and Nakamura (Chap. (A-2) and (A-17). ic^t i^x — viM — v[ v[n (A-17) (A-18) where o. special I{x) = f — G(f)e^'^' df (A-16) we have. For al/l3l = 3 we obtain (A-22) const = — [uoX + m(l — Mo)^ + w(3 — ?4)'] . - f)^ + in{kl. a pole of F(^) occurs indefinitely near a saddle point or of steepest descent. r) = const = = lm{itx + Uo{s im(kl. by Newlands (Chap. h. that Eq. Ref. The saddle point fo is therefore on the Now a{k. t) = const represents a line of steepest f a{k. 2. in branch points. 28). /'"(fo)?^ 0. 32). Ref. Emde [3] discussed the case where /"(fo) = 0.ff} is (A-21) The time factor being omitted again. and f = — ic)/ai. and by Eqs. if we put root for the typical case ai//3? line of descent. (A-20) has a single real = 3. taking into account the change notations. Writing such an integral in a form slightly different from (A-1). and possibly depth H oi a layer. The two methods are nearly equivalent since the paths pass through the saddle points at an angle 7r/4 to each other and can be deformed each into the other. Application of the method of steepest descent to the approximate evaluation of integrals met in problems on wave propagation was discussed by Nakano (Chap. For a further discussion of the connection between the methods see Eckart If [2]. Ott. Jeffreys: "Methods of Mathematical Physics. and p. Emde. 393-404. 300 = /c + ir in Eq. 49-55. 42." p.r. for example] that close to the saddle point the curve t) = const coincides with a parabola. 19.. Phys. these loops correspond to definite wave not necessary to know the detailed form of the line of steepest descent to determine which loops occur. 5. IV. It is contour in Sec. (A-23j |f| Thus the portions of tho curve (A-21) which correspond to large are straight lines in the fourth and third quadrants. pp. 399-417. C: The Approximate Solution of One-dimensional Wave Equations. 535-558. 473. 2. pp. London. 1948. 20. Watson.. and the critical distances ing pulses) first may be derived where loops (and the correspond- appear. Kelvin. pp. vol. Ann. the original contour. we obtain /ca: ± (m + n)T = respectively. S. . Ubertr. pp. F. vol. This approximation suggests that the curve of steepest descent If is of roughly parabolic shape. and B.: Die Sattelpunktsmethode in der Umgebung eines Pols. and jS. 1946. elek. pp. Talern. 2-5. if f varies on the curve a = const where p has its largest value. (A-6). For given values of m.. It may be necessary to distort the line of steepest descent to avoid crossing a branch line or a pole by inserting loops. Cam- bridge University Press. G. Physik. Cambridge Phil. 7. in Water of Any Depth. By of fo. p. vol. H. 1908. REFERENCES 1. Rev. Roy. 32). 211-214. vol. vol. Mod. or Math. Debjê. 2. a. Soc. the numbers and location of intersections depend on . only the number and location of intersections with the line of branch points are needed. 1887. 2.METHOD OF STEEPEST DESCENT Substituting f respectively. 67. Paper. Math. Proc. Phys. Eq. Arch.: The Limits of AppUcability of the Principle of Stationary Phase. and Zur Passmethode bei Passen mit Eckart. 214-218. Jeffreys. drei 4. cr(/c. 303. N. 1948. From this condition it may be derived [see Newlands (Chap. then the major contribution occurs in the vicinity of the saddle point and is given by (A-13). can be distorted to the path of steepest descent by arcs which contribute nothing.: Zur Passmethode. vol. Soc. 5. 6.. pp. 1943. SO. 1910. usually the real axis. (London). Lord: On or in a Dispersive the Waves Produced bj^ a Single Impulse Medium.: Naherungsformeln fiir die Zylinderfunktionen fiir grosse Werte des Index. (A-21) and neglecting k^Jf and /c^j/r*. Ann. 3. we obtain in the neighborhofxJ > p - po = ^ (f - fo)7"(ro) +^ (f - ro)V'"(ro) + • • • â-24) Ref. As was the case with the Sommerfeld types. n. Proc. P. 1918. H. it the kinetic and potential energies of small oscillations about can be expressed in the form • • T = |ang(^ ^hiiql W where = + ittaagf + + I62292 + • • • • + a^2q'iq2 + + 6i2?ig2 + • • • /-g_^N • • • a. = hh. are constants. b. and 6.. .- are the normal (dT\ dT _ dq.! The variables and the equations of motion d_ ^. new variables g.. = (B-4) = Ai cos (coit — €i) (B-5) of show that the periods of these "natural vibrations" or "normal modes vibration" are given by COi = — = -\/~ (-D-o) fA negative value of a. [Rayleigh (Chap.q\ + âzgf + + hh2ql + • • • /g_2N • • • coordinates. number of degrees of freedom are chosen in such a way that the values = determine a configuration of stable equilibrium. If Ref. (B-4) g. 6. may be introduced to yield the equations T = âiQi" W where a. Since a homogeneous quadratic function can be reduced by a linear transformation to the sum of squares.q.. 45)]. ^^ ^^ take the form aS' + The solutions of Eq. bi corresponds to instability with respect to the coordinate î- 370 .„_„.APPENDIX B RAYLEIGH'S PRINCIPLE Rayleigh's principle has been used for approximate computation of frequencies of vibrating systems with is much success. Its fundamental idea derived from the theory of small oscillations of a conservative system 1. are positive constants. dt \dq'J ~ dW dq. q^ the Lagrangian coordinates of a system with a finite g.. be no upper boundary for frequencies in continuous plates or strings). and this the mean values of the kinetic and potential energies. Thus the ampHtudes normal modes.B. W Let q vary as cos a)t = + + a. Assume that Then. say of the least frequency. is the smallest or the greatest ratio and I. It Love waves in 0) overhang a is assumed that given by w = V cos kx sin wt = kc (B-11) . = B. that the mean values of the kinetic and potential energies in a simple harmonic motion are equal and that they are T„ = co'Towhere Tq and r sin' cot dt W^ = W.Bl+ • • • h" (B_g) •••]r/ or sin If we now take into account." the closer minimum. By amplitudes of Eq. the last factors being equal to ^ each. as well as their frequencies. therefore. the frequency for vibrations of the system constrained to have certain amplitudes Bi. using the inequahties ^ ajzi.Bl+ h.g. To find an approximate value. "by observation or intuition.i3f co/. one has to compute. It concerns the approximate determination of the dispersion of a system composed of a homogeneous layer ( — if < 2 < heterogeneous the transverse medium wave is V (0 < 2 < co) (see Sec. (B-7) by a suitable constraint the system has only one degree if we put 9. (B-8).q the expressions for of freedom. for i 9^ exist. in case of a finite number of degrees of freedom). respectively.UAYLEIGIl's PKIXCIi'LE 371 of the where t^ is the period.-- f cos' o:t dt (B-9) T Jo T Jo Wq are the factors in brackets in Eqs.. and his example will now be discussed. and it cannot be less than the fundamental frequency (or larger than the greatest. (B-10) the frequency of a constrained vibration depends on all normal modes. T and T = W become h[a. we obtain w = = — ki ^2^-. are independent of one another. For these systems the theory requires the use of integral equations. |[6. 7-3). The was made by application of the Rayleigh principle to surface-wave propagation Jeffreys [2]. there systems (e. Temple [5] has justified the appHcation of will a? be to the true to con- this method tinuous systems. one can iihmediatel}^ will see that the limits colio and coL^ However. is done by equating The more accurately are given 5. ^2— j (±5-10) On assuming that hjai 6. r^ = ir/k and — 1 Z"^' / cos^ kx dx = TT 1 r" I cos^ ^ d^ 1 = - (B-14) Tx Jo Jo ^ The strain energy per unit volume [Love (Chap. Assume that the z/s)^. If t. 2W = y. for example. n' and s being constants.{el + ) el. therefore. = dv dx and.) dV ^^ cos^ kx sin^ wt + k'^V'^ sin^ kx sin^ wt (B-15) Hence the mean value with regard to x and / is given by «^''»=r„{(fj+^'''\ As a trial dz (B-16) form for V. 34. rigidity in the lower layer varies as fi'(l satisfy the equation of + . 102)] is reduced to a simple form for Love waves. p. we write the mean value of kinetic energy with regard to x and (B-12) it is easy to see that it becomes (B-13) J -H since. (B-11) differ from zero. They are (see Sec. Ref.'2 _ 1 - forms satisfy the boundary conditions but (B-18) does not motion for the bottom layer. 1. Jeffreys takes that corresponding to two uniform media (see Sec. since only two components of strain determined by Eq. 1-1) ^"' _ ~ ^ dz Br. 4-5) V = A sec {kHf) cos [k{H + z)y] ior -H <z <0 < 2 (B-17) (B-18) V = where These trial Ae'*^'^ for < 00 7 =^-1 .372 ELASTIC WAVES IN LAYEEED MEDIA T^ is where the amphtude a function of z to be found. } (B-20) As in the case of Eq. my) = ^ + "^'^ (1 + ^)} by (B-22) Jeffreys The [1]. to have a method to estimate the error and also a lower bound. After a simplification we obtain the approximate form of the period equation as ta.7/ r2/://^/)| (B-19) 8lf„ - ^ mA'/c'scc' (/c/7f)i(l + r)// + -^TZ^r^sin 2/cf + the 1 .Â^/. (B-10). this equation becomes m(1 + 7 ) sec' ikHi) H+ mn{2kHy) 2ky + ^ ('--") = IX see' {kHi)<(l + f)H + ^^ sin (2kHi) + of Eq.3'^ = mVp'. A simple presentation method is given by Timoshenko (Chap. as for the frequencies of higher . Jeffreys also applied this An method to the problem of Rayleigh is waves in layered media./ p and . however. an upper bound for a normal mode of an oscillating system can be determined by the Ritz-Rayleigh method. As is well known. the latter satisfy the equations of motion and the condition of continuity of stress at the interfaces. we obtain an equation for w or c by equating mean values of the kinetic and potential energies. + ^^^-. A further development of Rayleigh's method was given by Ritz [4]. If we use the velocities of shear waves given by . Ref. (B-21) follows: "' Hf {1 (l + i/ + i?F. The Ritz method of this yields a better approximation for the first mode as well modes of vibrations. It is important. 85). (B-22) as far as the terms in 1/s. and it may be proved that the value of unaltered to the if order of small increments in the expressions for displacements. exact period equation in this problem was also obtained asymptotic expansion of the exact equation agrees with Eq. 6.1.3^ = fj. 8r„ = ôi''A'ipêcUkHi) »'" I i^klH) 2/C7 + /.^(1 + .rayleigh's principle 373 Then the integrals (B-12) and (B-16) take the form ji . Such a method was developed by Temple [7] and Kato [3].'^)[l^ + -.--) <B-21) In this equation the variables 7 and 7' involve c. In this case an equation for c of the obtained as a function c is wave number first k. Bickley: "Rayleigh's Principle and Its Apphcations to Engineering.. 6. vol.. 334-339. Soc: Jeffreys. Monthly Notices Roy. 119. Temple. G. Temple. 276-293. vol. Suppl. 1909. 1952. Soc. 2. Astron. vol. 101-111. Jeffreys. G. Roy. J. (London). 4.: The Accuracy of Rayleigh's Method of Calculating the Natural Frequencies of Vibrating Systems. T. Proc.: On the Upper and Lower Bounds of Eigenvalues. pp.374 ELASTIC WA\rES IN LAYERED MEDIA REFERENCES 1. 1949. angew. vol. pp. Suppl.: Theory of Rayleigh's Principle Applied to Continuous Systems. Soc. G. Kato. pp. 2." Oxford University Press. Phys. 1-61. pp. 4. 7. 253-261. 204-224.: tJber eine neue Methode zur Losung gewisser Variationsprobleme der mathematischen Physik. Japan. A.: 3. London. 1934. H. vol. J. 3. Temple. The Effect on Love Waves of Heterogeneity in the Lower Layer. H. (London). Soc. pp. vol. Monthly Notices Roy. . pp. 1933. Proc. G. 211. A. Math. and W. 1928. Roy. Astron. 1928.. W. Geophys. Soc: Geophys.: The Surface Waves of Earthquakes. 135. 5. Ritz. reine u. M. 215 Benioff. 274-276 188. 285.. 265. 7.. 314-322 Birch. R. 16. T. F. M. G. 360 Adler. A. 115. M. G. T. 134 Cherry. F. 74. B.. 24. A. rigid. P. A... P. E.. 335 Crary. 171. F. transmission. 178. S.. 186 Carson. 10. 93 of emergence.. 359 Bakaliajev. 342 Christopherson. 245 Boundary. S. D. R. L. J. 96 Cole. 217. critical. 15 Brekhovskikh. 79. A. 208. 224 Bohnenblust.. 196-198 Bromwich. F. 184 Byerly. 13 Bancroft. 297-301 Crary waves. P. 115-119. R. 15. 324 Bleich. W... B. H. 163. 10. B&th. 113. R.. 105. 174. H.. J. R.. 311. H. E. J. H. 149. D. 157. 186 W. 116 Cauchy problem. of. D. 187.. 20 Blake. L. 342 Airy phase. 93 Blunt pulses. Bickley. 360 Cubical dilatation. 1G5. F. complex. A. 17.. 278 Caloi. C. 293 Critical angle. Bateman.. of incompressibility... 338 Brilliant. 12. 70 Anderson. 20.. O. 87. 45 Ansel. 198. D. 341 Cagniard. L.. Carlson.. 196 Carder... 20...... H. 219 Coulomb. S. 70. 324 Blake. condition 140 Contour integration. L. 292... 75 Boundary conditions. 143 Bergmann. 277 Bishop. 341 Baykal. 334. 132.. 31. K. 288 Carrier. 210. 7 Aeolotropic media. 107. 10. H. 189. P. 154. K. P. 104. effect of. J. FA. 308. D. 335 Angle. Barnes. 27 of incidence. 293 Arons. E. 44. J.. G. 1. 325 Blokhintzev. 216. 28. R. B. 15 Compressional waves. G.. E.. 91-93 Atchison. 324 Bontchkovsky. Chree.. 145.. 204 375 . 15 16. E. 324 S.. 278 Burg.. H. 205. 238 324 6 Coefficient.. 245 Booker. 266 Brown. 358 Aichi.. 330 Bernard. 87. R. 121 C. Blechschmidt. 25 Angular frequency. C. 323 Bullen. 358 Anderson. 334. 299 Cremer. 223.. 272 Conical waves. E. 181. Biot. 318 Constructive interference.. 169.. L... 196. D. 324 163.. R.INDEX Abeles.. L. G. 263. 183. K. dispersion. 2 Curvature. 169. V. 218. 98-104 Cooper. 374 M. 324 Clark. 118. 257259. 313 Banerji. 263 Curve. G. A. F. 132. K. 114 Antisymmetric vibrations.. 238. . C. 33 Dinger.. R.. 338-341. G. L. 196. 288 Faran. 176. 323 Gravity. P. W. internal. P. 325 Friction. H. 45 cutoff. 205. J. 310-313 W. 314 Cylindrical rod. S. 10 Herrick. D... 238 Fortier. 188 Donnell. 67 Hamilton. E. 36 INDEX Feshbach.. Y.. N. 64. P. 172.. 171.. 102... M. Energy Energy F. 211.. 312- 314 Czerlinsky. 324 Das Gupta. 324 Gilmore. E. 218 Ewing.. 355 M. 10. 2 Fu. 114 Dobrin. 121. 368 28 29-31. E. 288.. 327 Gronwall. 70. 247 Fisher. K. 299. Epstein. Deacon.. G. 157.. 324 Excitation amplitude. 255 Grazing incidence. 285. 68. 87-89. 82-89 Ground roll. 368 Emde. 305 Cylindrical tube. E.. 41 Forsterling. J. L. 169.. 87 Gans. P.. 174. M. 203. V. 293 of. 184. H.. 324 Goodier. M. K. 121 Frequency. 65. 246 Harrison. 285 Goland. 335 Gabriel.. 355 Haug. 341. complex. W. 187 Flexural vibrations. 272 Disturbance. J. 187 Dispersion curve. 278 Duvall. E.. J.376 Cutoff frequency. Hadamard. A. 229 partition. 204. K.. 236 Group velocity. P. H. angular. H. A. 181 Haskell. J. 322 H. A. 288.. cubical. A... 154. 238. J. 189. 217-219.. H. 284.... 238. M. C. effect Eckart. S.. 204 Displacement components. 151. ratio.. 324 Dyk. 149 Duhem. F.. O. M.. 208. 314 Fay.. 187. 222. H. 122 Heins. 365. 359 341 Dilatation. G.. J. A.. 67 Dix. G... 297. L. 323 Fliigge. 186 Ghosh. S. 236. 121 Gherzi.. 149. 81-86 Evernden. 13 Heelan. 196-198. E. F. traveling. V. 278. E. 365 De Juhasz. 187.. L. R.. J. C. 15 Duwez.. 187. 231. C. 296 Hartig. 323 Hencky.. 324 Dengler. B. 33. M. 349 Davies. S.. 113.. 71 Frohlich. 361 Gerjuoy. M. G. 144 Gutenberg. D. R. 293. C. E.. R. 292. 151. Fragstein.. 288. J. 319 Cylindrical waves.. 237 Donn. R. 139 Cylindrical hole. 31 Gotz.. 335-338. Gogoladze. 30 Griffis.. 21 Dietz. 263. S. 301. 287. E. 139 Frey. 7. R. 301. G. 278 Hart.. V. 324 Diaz. 278 Drake.. N. 8-10 Distortional waves. 285. 200-203. 188 Giebe. H. 126. C. 293. C. A.. 272 Friedlander. 361 Ergin. 200. 1 Displacement potentials. 188. T. R. v. 186 Debye. 311 Flexural waves. 29-31.. E. K. 187.. B.. 341 .. 165. 281.. 146 Excitation function. 263. 16 Hallen. 324 Forced waves. B. 342. P.. G.. 6 H. 17. M.. Lord. E. 212. 30. 341 Integration. T. 144. 283.. Interference.. 210. 194 . 127 London.. 322. 278 Kirchhoff's formula. 235 Lee. B. W. G. 371 174. 187-189. v. 21 A. H. Hubert.. 312. C. 13. 246 K. 164.. constructive. 306 W. 70.. Imperfectly elastic media..INDEX Herrmann. liquid-liquid. 54. H. D. G.n. T. 36. 268 Jeffreys. 113. N. 204-207. 56. 187 368 Knopoff. 325 Leet. E. D. 288 Kdrmd. 296. M. F. 111. 323 S.. 227. 15.. 120 Hudson. 70 Kawasumi. 306 Longuet-Higgins. 74. 193. 18. 323 Ingard. B. S.. 50. 377 A. 325 232.. Levine.. 272. 44. 62-70. 288. 238... R. J.. coefficient of.. Jeans. Kahan. Koppe. 350. Hillier. W.. 259.. J.. A.. 335 Kruger. Lieber. 13. 342. 265. S. 249 194 Kelvin. 279 Landisman. 279 Kato. 250 Kaufman. W.. S. 193. 21 Kinsler. 198. A. 319. A. 69. 279 Lg. U. 274. 373 Katz. 61. 194 vibrations. D. solid-solid. 325 Leaking modes. T. 44. 219 367 215. M. 272. T.. 78 liquid-solid.. 167. L. 210. 175. 325 Huygens' principle.. J. 279 Jeffreys.. E. 322. 56. C. 223 79 at. 270. 342 319. 211. 324 King. H. J. 325 Junger. V. K. 16 Hirono. 324 Homma. 112 Krasnushkin. 194.. 34. 187. E. 16 Lamb. 189. H. 157. Lampariello. 184 272 Lebedev. 69 Inouye. 257. 98-104 Interface. Kanai. A. 234. 347. B. S. 341 Lapwood.. contour. 135. 74 Kolsky.... H. across continents. G.. 325 Lurye. 365. N. 34. 266 Joos. T. 93. 29. 47. P. 33. 115 Kupradze.. M. 94 Liquid bottom. 362 36.. V. 186... 325 Lee.. 365. N. R.. 34. E. 205. K.. 238 Line source. Longitudinal vibrations. 70 Knott. 156.. H. 187 Love. W. 222. Jardetzky. 341.. 319. 121. Keller. 350 Kane. 341 Jobert. F. 247 Kling.... L. G. 274. B. Lavrent'ev. Love waves. P. Jacobi. 269.. 192... 71 Lamb's problem. L. 238. 325 Hersey. 232. 246 15... 216 Lunts. 358 Holden. 213 across oceans.. 97. 325 Huter. 34. 34. Rg waves. 214. 229. 83 114 Lawrence. 194. S.. H. E. von. S. 371-373 Lindsay. 200-203 waves generated Internal friction.. 205. 281. 135. 313 Hughes. D. L. 9. 337 114. W. 112. Honda. 348 Lewy. 367. J. H... 15 Keilis-Borok. 123 269. 87. 367 Ml M% vibrations. M. Y. 245. 121. 10 I.. 225. E.. L. 346.. 124. 140 Lazutkin.. H. 49 Linehan. L. 7 Incompressibility. R.. 105. 10. 293. 7. 111. 198. F. 154. A. 341. 142 Nishimura. T. 349 Meres. 151.. 21. normal. K. 365. J. 266 Matuzawa. Lord. 278. 250 Poritzky. 301. T. D. Y.378 MacDonald. M. Y. 132.. A. 196-198. 266 Poisson constant.. W. 287. H. 157.. 69. 326 13. H.. 172.. 70.. 87. 31. 144-155. 184. M. W. 60.. 278 Marcelli. 21. R. 67 Phase change. 312-314 O'Neil. W. K. 22 Niessen. 368 Pearson. 172. 21 Macelwane... 104. 222. M. symmetric. 368. 126.. 174. R. L. H. Point source. E. 93. F. C. 288. A. M. 288. 165. 72 Plane waves.. H. 32. 248 Nolle. E. 370 Northwood. 224. 138. K. N.. 115.. R. 26. 131. A. W. 35. 326 Pekeris. 188 Muskat. K. 238. 249 Mackenzie. 127. R. H. 292 325 Mindlin. M. 111... 236. 359 McLachlan.. 343. 144. 232. C. 260 Milne.. 223. R. G. 15 Potentials. 276. 351. 10 Murphy. 288 Mode. 224 for). 341. 199. 310. L. 244. 70. A. antisymmetric. N. D. D. Jr. 279 Officer. 34.. 97. 87 INDEX Oestreicher.. 61. 192.. L. 281. 335 Murnaghan. 185-189.. 104 Radiation condition. 6 Poisson solution.. 333. L. 183. of stationary phase. 22 Petrashen. 326. 121 National Research Council. 368 Nakano. 121 L. H.. 105.. 325 Mindlin. M. 15.. 87. 156 Oldham. 104 Press. J. 325 Mantle waves. M. 176. 288 Nishizawa. W. 325 Mainka. 279 Matumoto. 102. 369 Newton. 238. 162. J.. M. 222. 217- 219. 175. 71 McLemore. A. J. 213. 342.. H.. 123 Miche. displacement. E. 249 Neumark. 293. 102. 355 Pulse (generalization blunt.. D. 70 Norton. Mims. 114. 238. H.. 287. J. 370 Mohorovicic. 367 of steepest descent. 250 Rayleigh. 156. 276 Meissner. E. 67. 297. 94. 189.. H. 330. 187. 66. A. B.. W. 279 Normal mode.. 279. 60. 279 Nomura. 358 Maxwell solid. 22 Malvern. 69 Oliver. J. 8-10 velocity. K. 281 Nakamura. B.. 293.. G. 292.. 49. 347. A. 113. 362 Pike. 293 Pochhammer. 20. 326 Ramirez. L.. 22 Pinney. 278 Mattice. L. 13 Rakhmatulin. R. 54. D. 334 Pendse. G... W. C.. 123 Method. 246 McSkimin... 266.. 198. E.. M. M. 217-219. 22. C. 368 Naryskina. P. T. 81 Phase velocity. 338 Rayleigh equation.. F. G.. 12 Prescott. 221 Morse. H. T. 275 Rayleigh principle. G. H. 365 Picht. 68 Pich. 169. 64.. 142.. 138. 211. J. 9 Penney. E. 129.. M.. 172. 164. 225. R. 64-66. 362 Martin. 16 Pondrom. 196. 216 Marsh.. C.... E. 194 Newlands.. 8. 325 Pontificia Academia Scientiarum.. 10 Plate vibrations. 19. 113. J. F. 279 Osborne. 296 Ott. 353. 342. 370 . 38.. W. 56. 265.... 22 Morse. 216. 186 Microseisms. L.. air-('Oupled. 263 Schuster.. 163. A. 73 Stokes. GO. 277 49 Rider waves. 326 Ritz. 142. K.. 358 Sakai. Swanson. 15. E. E. 66. 115 Schoch. K.. 327 W. 266 Spherical waves. W. 19. 80. F. D. 111-113. 213. 3 269.. 214. G. O. 93. 358. C. 365 Sakadi. 250 . N. 10.. 362 Roseau... 360 Sauter. M. 49. 350 S. 326 Sharpo. 319 Strain components. 57... 230. 67. 359 Sack. A. E. 280 Roy. T.. 156. 15 Roever. O. 292 262. G. K. 111. 123 Schelkunoff. 288 of.. 279 Takayama. 161. 15. 137. 15 Shapiro. N. 10. 12 Sheehy. 213. 104 Rydbeck. 107. 67. 196. 274.. 365 Source.. 183 Rinehart. 164.. Z. 229.. 131. Sezawa.. 196 free. 115. W.. 327 Somigliana. 192. 87 r-phase. method Steepest descent. 373 236 47 across continents. L. 163. 121 Reflection coefficient. 353. 188 Simon. 67. 330 Refraction arrival.. 90 Slichter. T. 38. C. 274. 288 Stulken. 353 suboceanic. G. 225. 222. 227. 33.. 342. 200-203 beyond critical angle. A. L. 28. 205. 124. 169. P. O. 199. 362 Rzhevkin.. Sobolev.. 187-189. 281. Stoneley waves. 350... V. 278. 70 Rohrbach. A. S. 67 mantle. 210... 257. H. 20.. 65 Sato. 67... 205. 69 Reflection.INDEX Ruyleigh waves. 38. S. 184 Sutton. 121. 266. 260. 263 velocity of. 344. W. G. R. 335 Somers. 342. 22 Schermann. J. A.. 293. 351. 13. 56. 359 Saddle point. 297. 263-266. J.. A. 113 Refraction index. 31.. H. 64. L. 373 point. 202. B. 64.. 196. L. 120. 156.. C. 319.. at critical angles. 268. 341 Schwinger. 173. 166. 194.. 156. 114 Reichel. J.. R. 96 S. 24.. 348.. O. T.. 54. 93. 197. 94. 241 Schaefer. 104 Refraction waves. 10. 208. H. Y. 33. Ricker. 34.. G. J. 29 Shear waves. method Stenzel. E. C. 112. 367 365 Stewart. Shatashvili.. 172-174. K. J. 218. G. 280 Rudnick. Swainson. 10. 341 Shurbet. 25 Smirnov. 325 Selberg. 196 Sound channel. M. 13. 163. 293 Scholte.. 156. 273. F. 12 Stanford. 49 Rebeur-Paschwitz. 2 Stratton. I. 194. H. J. 1... 288 SOFAR. 113. 110-113. F.. 104. Sykes. 15 Takaku. J. 230. 349. 112. 335 E. 73 Sommerfeld. 324 Schmidt. S. of. 247 Suzuki. 345. L. 13 Stress components. 94.. A... 210. G3-70. I. S. 183. 194. 113. Stationary phase. 288. 194. D. 198. 186. 251 Reissner. 178. G. 337 line. 254 20. J. 55. H.. R. Richter. 194. 253 Stoneley. C. 358 164. L... 40.. 283. R. V.. 341 Shurbet. 87 at free surface. 238.. 278 Symmetric vibrations.. L. 69 Stone.. 144. 178. 379 Serigbush. 194. E. 65. 27. R. 87.. F. 194 Ma. 284-286 Stoneley. D. 363 Weinstein. A. 174. 23 Whittaker. 330 Young's modulus. 341 Wuster.. 238.. S. A.. G. 203.. 23 Weyl. 238 Thornburgh. G. 183 {See also Velocity) Van Van Cittert. 165. 121 . 6 Wadati. 126. M. 194. E. 229. 311 S. 327 Zvolinskii. 293 symmetric. H. compressional. N. 342 Rayleigh waves. 118.. 144 phase. 329. 10 J. 238 der Pol. F. V... Vanek.. P. 272 Yennie.. 115 Whipple. H. 13. 12 Vibrations.. 178. 123 Wolf. E. R. 41 generated at an interface. 327 Woollard. 361 Vine. 272 318 Crary. S.. 79. 210. G.. J. H. conical.380 Tamarkin. Uller. 319 water. 285. 36 distortional. of compressional waves. 297.. 49 8.. R. Tatel. P. of shear waves. B. 299 Tillotson.. J. 10. 176. L. O. D. I. K. 186 Usami. 8. antisymmetric. T. 327 Whitham. 285. 187 White. INDEX 314 70 280 Watson. 187-189. Taylor. Love (see Love waves) 195.. 292. coefficient. H. T. W. L. 10. 293. 292 torsional.. 195. 26. E. 341 Lg. 335-338. 263... 107. 219 Torsional vibrations. B. 295 Thompson... 12 spherical. 361 Worzel. G. D. A. 118-119 Wave length. C. 199. 280 Thomson.. 14.. 373 Teltow.. J. 27. I.. 194. 33. P.. A. 23 Velocity. M.. 205. 272 Yosiyama. 254 H. 301. C. P.. 278 plane. 15. {see Rayleigh waves) Traveling disturbance. 194. 96. H. 361 Viscosity... 254 Usdin. W. 111-113. 230 Timoshenko. E. M. 11 Wave number.. 198. 114 Thorne.... 163. 196. F. N. 373 Terman. Wood. 194 plate. C. 306 Wolf. 67 refraction.. 238.. flexural. 293 311 longitudinal. B. 312-314 forced. J. 135 Wiechert. E. 215-217.. Rg. 205. 28 Walters. 11 group. J. 114 183 Uflyand.. W. 327 Rayleigh rider. 121 Zener.. A. 12.. 312. 299 cylindrical. 169. 367 Tartakovskii. E. 114 Tolstoy. 188. 186 Wilson. 94 Temple. 73 Water waves. G. J. P. 11 Velocity potential. 311 Transmission Trauter. 12 329 Ursell. G. H. 10. C.. 104. 68. T. 178-180.. 205.. 284-286. 325 Ml. 285. 104 J.. 287. 91-93 Voigt solid.. 212. 96 mantle. 183 Zaicev. 208. Wave equation. K. 283.. 341 Walker. 288.. 272 flexural. 288 shear. Y. 9 Wave fronts. 283. P. 68 of Weber. 358 White. G. effect of. 13 Waves. . .Date Due Due^.^* Returned Due Returned MAV 3 1 '^^-: mTT^ ^ 2 fi m m^ APR 2 s v^rA [ 1S92 m. '' ^ 2 % ' APR 12 ' . SClENGEt Elastic waves in layered m UGE 550.l89 3 lEbS D11D3 T71S .82 C726C. no. 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