CONTENTS

Preface V
List of Symbols .... . xi
1. Fundamental Equations and Solutions .... 1
1-1. Equations of Motion .... 1
The Equation of Continuity . ... 5
1-2. Elastic Media . . . . 5
Isotropic Elastic Solid 5
Ideal Fluid . . . . 6
1-3. Imperfectly Elastic Media 7
1-4. Boundary Conditions 7
1-5. Reduction to Wave Equations 7
1-6. Solutions of the Wave Equation 10
Plane Waves 10
Spherical Waves 12
Poisson and Kirchhoff Solutions 16
General Solution of Wave Equation 17
Some Special Forms Used 18
References .... 20
2 Homogeneous and Isotropic Half Space 24
2-1 Reflection of Plane Waves at a Free Surface 24
Incident P Waves 26
Incident S Waves 28
Partition of Reflected Energy 28
Reflection at Critical Angles 29
2-2. Free Rayleigh Waves 31
2-3. Integral Solutions for a Line Source .... 34
Surface Source . . ... 34
Internal Source . ... 36
2-4 Integral Solutions for a Point Source 38
Surface Source 38
Internal Source 42
2-5. Evaluation of Integral Solutions 44
Application of Contour Integration 44
Residues 47
Branch Line Integrals: Line Source 49
Application of the Method of Steepest Descent 59
2-6. Generalization for an Arbitrary Time Variation 61
2-7. Other Investigations ... . . 64
2-8 Traveling Disturbance . 67
2-9 Experimental Study of Lamb's Problem . . 69
References . 71
3. Two Semi-infinite Media in Contact 74
3-1. Reflection and Refraction of Plane Waves at an Interface . . 74
Rigid Boundary . . . 74
General Equations . . 76
Liquid-Liquid Interface .78
Liquid-Solid Interface 79
Solid-Solid Interface 83
3-2. Reflection of a Pulse Incident beyond the Critical Angle 93
3-3. Propagation in Two Semi-infinite Media: Point Source . 93
Two Liquids . 94
Fluid and Solid Half Spaces . . 105
Two Solids . . 107
Stoneley Waves . Ill
3-4. Further Remarks on Waves Generated at an Interface 113
3-5. Other Investigations . 115
References 121
4. A Layered Half Space . . 124
4-1 General Equations for an n-Layered Elastic Half Space . 124
4-2. Two-layered Liquid Half Space 126
Discussion of Solutions . . . . 137
Generalization for a Pulse . 142
4-3. Three-layered Liquid Half Space 151
4—1. Liquid Layer on a Solid Bottom 156
Compressional-wave Source in the Solid Substratum 157
Suboceanic Rayleigh Waves. First Mode 166
Compressional-wave Source in the Liquid Layer 174
Leaking Modes . 184
Some Aspects of Microseisms . . 185
4-5. Solid Layer over Solid Half Space 189
Rayleigh Waves General Discussion . ... 189
Propagation of Rayleigh Waves across Continents 196
Ground Roll 200
Theoretical Rayleigh-wave Dispersion Curves 204
Love Waves' General Discussion 205
Love Waves across Continents 213
Love Waves across Oceans 216
Lg and Rg Waves 219
Other Investigations 222
4-6. Three-layered Half Space 224
Oceanic Rayleigh Waves with Layered Substratum 225
Love Waves 227
4—7. Air-coupled Rayleigh Waves 230
Air-coupled Ground Roll 236
4-8. Remarks Concerning the Problem of an n-Layered Half Space .... 238
References . 245
5. The Effects of Gravity, Curvature, and Viscosity 255
6-1. Gravity Terms in General Equations 255
5-2. Effect of Gravity on Surface Waves 257
Rayleigh Waves- Incompressible Half Space 257
Gravitating and Compressible Liquid Layer over Solid Half Space . . . 260
5-3. Effect of Curvature on Surface Waves ... 263
Cylindrical Curvature 263
Sperical Curvature 265
5-4. General Solutions for a Spherical Body ... 266
Wave Propagation in a Gravitating Compressible Planet 269
5-5. The Effect of Internal Fiiction . . . . 272
Voigt Solid 272
Maxwell Solid 276
Internal Friction in Earth Materials 277
References . . 278
6. Plates and Cylinders 281
6-1. Plate in a Vacuum 281
Symmetric Vibrations (Mi) 283
Antisymmetric Vibrations (Af2) 285
Interpretation in Terms of P and SV Waves 286
Impulsive Sources 286
Other Investigations 28S
6-2 Plate in a Liquid 288
Symmetric Vibrations 292
Antisymmetric Vibrations 293
Other Investigations 293
6-3 Floating Ice Sheet 293
SH Waves 293
SV and P Waves 295
Crary Waves 299
Flexural Waves from an Impulsive Source .... 301
6-4 Cylindrical Rod in a Vacuum 305
Longitudinal Vibrations 306
Torsional and Flexural Vibrations 311
6-5 Cylindrical Rod in a Liquid 314
6-6 Cylindrical Hole in an Infinite Solid 314
6-7 Liquid Cylinder in an Elastic Medium 317
6-8 Cylindrical Tube 319
References . 323
7 Wave Propagation in Media with Vabiablb Velocity 328
7-1. Wave Propagation in Heterogeneous Iaotropic Media 328
7-2. Sound Propagation in a Fluid Half Space . 330
SOFAR Propagation 335
The T Phase 341
7—3. Love Waves in Heterogeneous Isotropic Media '. 341
Homogeneous Layer over Heterogeneous Half Space: Matuzawa'a Case . 343
Sat6's Case 344
Jeffreys' Case 346
Meissner's Case 347
7-4. Rayleigh Waves in Heterogeneous Isotropic Media 349
Mantle Rayleigh Waves 353
7-5. Aeolotropic and Other Media 358
References 360
Appendix A. Method op Steepest Descent 365
References 369
Appendix B. Ratleiqh's Pbinciple 370
References 374
Index 375

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