Elastic Waves High Frequency Theory Monographs and Research Notes in Mathematics Series Editors: John A. Burns Thomas J. Tucker Miklos Bona Michael Ruzhansky Variational-Hemivariational Inequalities with Applications Mircea Sofonea, Stanislaw Migorski Optimization and Differentiation Simon Serovajsky Willmore Energy and Willmore Conjecture Magdalena D. Toda Nonlinear Reaction-Diffusion-Convection Lie and Conditional Symmetry, Exact Solutions and Their Applications Roman Cherniha, Mykola Serov, Oleksii Pliukhin Mathematical Modelling of Waves in Multi-Scale Structured Media Alexander B. Movchan, Natasha V. Movchan, Ian S. Jones, Daniel J. Colquitt Integration and Cubature Methods A Geomathematically Oriented Course Willi Freeden, Martin Gutting Actions and Invariants of Algebraic Groups, Second Edition Walter Ricardo Ferrer Santos, Alvaro Rittatore Lineability The Search for Linearity in Mathematics Richard M. Aron, Luis Bernal-Gonzalez, Daniel M. Pellegrino, Juan B. Seoane Sepulveda Difference Equations: Theory, Applications and Advanced Topics, Third Edition Ronald E. Mickens For more information about this series please visit: https://www.crcpress.com/Chapman--HallCRC-Monographs-and-Research- Notes-in-Mathematics/book-series/CRCMONRESNOT MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS Elastic Waves High Frequency Theory Vassily M. Babich St. Petersburg Department, Steklov Mathematical Institute, Russia Mathematical Faculty, St. Petersburg State University, Russia Aleksei P. Kiselev St. Petersburg Department, Steklov Mathematical Institute, Russia Physical Faculty, St. Petersburg State University, Russia Institute for Problems in Mechanical Engineering, St. Petersburg, Russia Translated from Russian by Irina A. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface xiii Introduction xv List of Basic Symbols xix 1 Basic Notions of Elastodynamics 1 1.1 Displacement, deformation, and stress . . . . . . . . . . . . . 1 1.1.1 Displacement vector and strain tensor . . . . . . . . . 1 1.1.2 Stress tensor . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Lagrangianapproach to mechanical systems . . . . . . . . . 4 1.3 Elastodynamics equations . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Kinetic and potential energies as quadratic functionals 7 1.3.2 Properties of elastic stiffnesses . . . . . . . . . . . . . 7 1.3.3 Derivation of elastodynamics equations . . . . . . . . 8 1.3.4 Navier and Lam´e operators . . . . . . . . . . . . . . . 10 1.4 Classical boundary conditions . . . . . . . . . . . . . . . . . 10 1.4.1 List of boundary conditions . . . . . . . . . . . . . . . 11 1.4.2 Hamilton principle and boundary conditions. . . . . . 13 1.5 Isotropic medium . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.1 Consequences of the invariance of with respect to W rotations . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.2 Consequences of the positive definiteness of . . . . 16 W 1.6 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7 Time-harmonic solutions . . . . . . . . . . . . . . . . . . . . 18 1.7.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . 18 1.7.2 Time-averaging . . . . . . . . . . . . . . . . . . . . . . 19 1.8 Reciprocity principle . . . . . . . . . . . . . . . . . . . . . . 20 1.8.1 The time-harmonic case . . . . . . . . . . . . . . . . . 21 1.8.2 Non-time-harmonic case . . . . . . . . . . . . . . . . . 22 1.9 ⋆ Comments to Chapter 1 . . . . . . . . . . . . . . . . . . . 22 References to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . 23 v vi Contents 2 Plane Waves 25 2.1 Plane-wave ansatz . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Phase velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Normal velocity of a moving surface . . . . . . . . . . 27 2.2.2 Phase velocity and slowness . . . . . . . . . . . . . . . 28 2.3 Plane waves in unbounded isotropic media . . . . . . . . . . 29 2.3.1 Eigenvalue problem . . . . . . . . . . . . . . . . . . . 29 2.3.2 Wave P . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Wave S . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.4 Time-harmonic waves P and S . . . . . . . . . . . . . 31 2.3.5 Polarization of time-harmonic waves P and S . . . . . 32 2.3.6 Group velocity . . . . . . . . . . . . . . . . . . . . . . 35 2.3.7 Energy relations for time-harmonic waves . . . . . . . 35 2.3.8 ⋆ Potentials . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Plane waves in unbounded anisotropic media . . . . . . . . . 39 2.4.1 Eigenvalue problem . . . . . . . . . . . . . . . . . . . 39 2.4.2 Phase velocity . . . . . . . . . . . . . . . . . . . . . . 40 2.4.3 Group velocity . . . . . . . . . . . . . . . . . . . . . . 40 2.4.4 Slowness, slowness surface, velocity surface . . . . . . 41 2.4.5 ⋆ Rayleigh principle . . . . . . . . . . . . . . . . . . . 43 2.5 ⋆ Local velocities and domain of influence . . . . . . . . . . . 43 2.5.1 Statement of the problem . . . . . . . . . . . . . . . . 44 2.5.2 Energy lemma . . . . . . . . . . . . . . . . . . . . . . 45 2.5.3 Uniqueness theorem . . . . . . . . . . . . . . . . . . . 49 2.6 Reflection of plane waves from a free boundary of an isotropic half-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.6.1 Upgoing and downgoing waves . . . . . . . . . . . . . 50 2.6.2 Waves of polarizations SH and P SV . . . . . . . . 51 − 2.6.3 The case of polarization SH . . . . . . . . . . . . . . . 51 2.6.4 The case of polarization P SV . . . . . . . . . . . . 53 − 2.6.5 Snell’s law . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.6.6 Total internal reflection . . . . . . . . . . . . . . . . . 57 2.6.7 Energy flow in an inhomogeneous wave P . . . . . . . 58 2.6.8 ⋆Energyflowunderreflectionfromaboundaryandthe unitarity of the reflection matrix . . . . . . . . . . . . 58 2.6.9 Reflection of non-time-harmonic waves . . . . . . . . . 61 2.7 Classical plane surface waves in isotropic media . . . . . . . 63 2.7.1 Classical Rayleigh wave . . . . . . . . . . . . . . . . . 64 2.7.2 Classical Love wave . . . . . . . . . . . . . . . . . . . 67 2.7.3 ⋆Thetotalinternalreflectionandconstructiveinterfer- ence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.8 Plane surface waves in isotropic layeredmedia . . . . . . . . 72 2.8.1 Waves P SV . . . . . . . . . . . . . . . . . . . . . . 73 − 2.8.2 Waves SH . . . . . . . . . . . . . . . . . . . . . . . . 73 Contents vii 2.9 Plane waves in arbitrarily layered media . . . . . . . . . . . . 74 2.9.1 Eigenvalue problem . . . . . . . . . . . . . . . . . . . 74 2.9.2 Virial theorem . . . . . . . . . . . . . . . . . . . . . . 76 2.9.3 Group velocity theorem . . . . . . . . . . . . . . . . . 78 2.10 ⋆ExistenceoftheRayleighwaveinananisotropichomogeneous half-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.10.1 One-dimensionalproblemandthecorrespondingenergy quadratic form . . . . . . . . . . . . . . . . . . . . . . 80 2.10.2 Inhomogeneous plane waves and the continuous spec- trum of the operator γ . . . . . . . . . . . . . . . . . . 81 2.10.3 Variational principle . . . . . . . . . . . . . . . . . . . 82 2.10.4 Discrete spectrum of γ . . . . . . . . . . . . . . . . . . 83 b 2.11 ⋆ Comments to Chapter 2 . . . . . . . . . . . . . . . . . . . 85 References to Chapter 2 . . . . . b. . . . . . . . . . . . . . . . . . . 87 3 PointSourcesandSphericalWavesinHomogeneousIsotropic Media 93 3.1 Delta functions . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.2 Scalar point source problems . . . . . . . . . . . . . . . . . . 97 3.2.1 Time-harmonic source . . . . . . . . . . . . . . . . . . 98 3.2.2 Determining a unique solution.Key ideaofthe limiting absorption principle . . . . . . . . . . . . . . . . . . . 99 3.2.3 Nonstationary source. . . . . . . . . . . . . . . . . . . 100 3.3 Point sources in a homogeneous, isotropic, elastic medium. Time-harmonic case . . . . . . . . . . . . . . . . . . . . . . . 102 3.3.1 ⋆ Center of expansion and center of rotation as limit problems for spherical emitters . . . . . . . . . . . . . 103 3.3.2 Concentrated force . . . . . . . . . . . . . . . . . . . . 106 3.4 Pointsourcesinahomogeneous,isotropic,elasticmedium.Non- stationary case . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.5 Conditions at infinity and uniqueness . . . . . . . . . . . . . 114 3.5.1 Limiting absorption principle . . . . . . . . . . . . . . 114 3.5.2 ⋆ Radiation conditions . . . . . . . . . . . . . . . . . . 115 3.5.3 Uniqueness theorem in the nonstationary case . . . . . 120 3.6 ⋆ Comments to Chapter 3 . . . . . . . . . . . . . . . . . . . 121 References to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 122 4 Ray Method for Volume Waves in Isotropic Media 125 4.1 Ray ansatz and transport equations . . . . . . . . . . . . . . 125 4.1.1 Ray ansatz and local plane waves . . . . . . . . . . . . 125 4.1.2 Recurrent system . . . . . . . . . . . . . . . . . . . . . 128 4.1.3 Waves P and S . . . . . . . . . . . . . . . . . . . . . . 129 4.2 Eikonal equation and rays . . . . . . . . . . . . . . . . . . . 130 viii Contents 4.2.1 Fermat functional and rays . . . . . . . . . . . . . . . 130 4.2.2 Solving the eikonal equation with the help of rays . . 131 4.2.3 Cauchy problem for the eikonal equation. . . . . . . . 132 4.2.4 Ray coordinates and field of rays . . . . . . . . . . . . 135 4.2.5 ⋆ Complex eikonal . . . . . . . . . . . . . . . . . . . . 136 4.3 Solving transport equations. The wave P . . . . . . . . . . . 138 4.3.1 Zeroth-orderapproximation.Consistencyconditionand the Umov equation . . . . . . . . . . . . . . . . . . . . 138 4.3.2 Zeroth-order approximation. Formulas for and uP0 . 140 E 4.3.3 Ray coordinates and the geometrical spreading . . . . 144 4.3.4 Anomalous polarization . . . . . . . . . . e. . . . . . . 147 4.3.5 ⋆ First longitudinally polarized correction . . . . . . . 148 4.3.6 ⋆ Higher-order approximations . . . . . . . . . . . . . 148 4.4 Solving transport equations. The wave S . . . . . . . . . . . 149 4.4.1 Zeroth-order approximation. A preliminary considera- tion. The Rytov law . . . . . . . . . . . . . . . . . . . 149 4.4.2 Rytov law. The case of complex uS0 . . . . . . . . . . 152 4.4.3 Anomalous polarization . . . . . . . . . . . . . . . . . 153 4.4.4 ⋆ First transversely polarized correction . . . . . . . . 154 4.4.5 ⋆ Higher-order approximations . . . . . . . . . . . . . 155 4.5 Reflection of the wave defined by a ray expansion . . . . . . 155 4.5.1 Ansatzandthestatementoftheproblemofdetermining reflected and converted waves . . . . . . . . . . . . . . 156 4.5.2 Constructing the wavefield in higher orders . . . . . . 158 4.6 ⋆ Riemannian geometry in ray theory . . . . . . . . . . . . . 160 4.6.1 Riemannian geometry and Fermat principle . . . . . . 160 4.6.2 Parallel translation in Riemannian metric and the Rytov law . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.7 Geometrical spreading in a homogeneous medium . . . . . . 165 4.7.1 Lines of curvature and Rodrigues’ formula . . . . . . . 166 4.7.2 Derivation of a formula for J . . . . . . . . . . . . . . 167 4.7.3 Onthevanishingofthegeometricalspreadingandcaus- tics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.8 ⋆ Geometrical spreading under reflection, transmission, and conversion in the planar case . . . . . . . . . . . . . . . . . . 168 4.8.1 Specific features of the planar case . . . . . . . . . . . 168 4.8.2 Jacobi equation and geometrical spreading. . . . . . . 170 4.8.3 Calculation of the initial data for the Jacobi equation in the case of monotype reflection . . . . . . . . . . . 172 4.8.4 Calculation of the initial data for the Jacobi equation for the case of reflection with conversion . . . . . . . . 176 4.8.5 Calculation of the initial data for the case of transmis- sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 4.8.6 Case of constant velocities . . . . . . . . . . . . . . . . 177 4.8.7 Focusing under reflection . . . . . . . . . . . . . . . . 179 Contents ix 4.9 ⋆ Nonstationary versions of the ray method . . . . . . . . . . 179 4.9.1 High-frequency asymptotics and asymptotics with re- spect to smoothness . . . . . . . . . . . . . . . . . . . 179 4.9.2 Other nonstationary versions . . . . . . . . . . . . . . 181 4.10 ⋆ Comments to Chapter 4 . . . . . . . . . . . . . . . . . . . 181 References to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 184 5 Ray Method for Volume Waves in Anisotropic Media 191 5.1 Recurrent system and eikonal equation . . . . . . . . . . . . 191 5.1.1 Recurrent system . . . . . . . . . . . . . . . . . . . . . 191 5.1.2 Eikonal equation . . . . . . . . . . . . . . . . . . . . . 192 5.2 Rays and wavefronts . . . . . . . . . . . . . . . . . . . . . . . 193 5.2.1 Cauchy problem for a nonlinear equation . . . . . . . 193 5.2.2 Characteristic system . . . . . . . . . . . . . . . . . . 194 5.2.3 Special case: eikonal equation . . . . . . . . . . . . . . 195 5.3 ⋆ Fermat principle and Finsler geometry . . . . . . . . . . . 196 5.3.1 Raysasextremalsofacertainfunctionalofthecalculus of variations. . . . . . . . . . . . . . . . . . . . . . . . 196 5.3.2 Finsler metric . . . . . . . . . . . . . . . . . . . . . . . 198 5.3.3 Fermat principle . . . . . . . . . . . . . . . . . . . . . 199 5.3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . 199 5.4 Solution of transport equation for u0 . . . . . . . . . . . . . 200 5.4.1 Consistency condition and the Umov equation . . . . 200 5.5 Higher-order terms . . . . . . . . . . . . . . . . . . . . . . . 201 5.6 ⋆ Comments to Chapter 5 . . . . . . . . . . . . . . . . . . . 203 References to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . 204 6 Point Sources in Inhomogeneous Isotropic Media. Wave S from a Center of Expansion. Wave P from a Center of Rota- tion 207 6.1 Statement of the problem and elementary consideration . . . 208 6.1.1 Statement of the problem . . . . . . . . . . . . . . . . 208 6.1.2 Non-applicability of ray formulas near the source point 209 6.1.3 Elementary locality approach . . . . . . . . . . . . . . 209 6.2 Structure of the wavefield near the source point in more detail 211 6.2.1 Recurrent system . . . . . . . . . . . . . . . . . . . . . 211 6.2.2 ⋆ On solving equations (6.25)–(6.27) . . . . . . . . . . 213 6.2.3 Intermediate zone . . . . . . . . . . . . . . . . . . . . 214 6.3 Preliminary notes on calculating diffraction coefficients χ1 for a center of expansion and ψ1 for a center of rotation . . . . . 214 6.3.1 Wavefield in the homogeneous-medium approximation 215 6.3.2 How to find χ1, or discussion of the matching procedure 215 6.3.3 The case of a center of rotation . . . . . . . . . . . . 216