Waves And Rays In Seismology: Answers To Unasked Questions

by Michael A Slawinski

Overview

The author dedicates this book to readers who are concerned with finding out the status of concepts, statements and hypotheses, and with clarifying and rearranging them in a logical order. It is thus not intended to teach tools and techniques of the trade, but to discuss the foundations on which seismology — and in a larger sense, the theory of wave propagation in solids — is built. A key question is: why and to what degree can a theory developed for an elastic continuum be used to investigate the propagation of waves in the Earth, which is neither a continuum nor fully elastic. But the scrutiny of the foundations goes much deeper: material symmetry, effective tensors, equivalent media; the influence (or, rather, the lack thereof) of gravitational and thermal effects and the rotation of the Earth, are discussed ab initio. The variational principles of Fermat and Hamilton and their consequences for the propagation of elastic waves, causality, Noether's theorem and its consequences on conservation of energy and conservation of linear momentum are but a few topics that are investigated in the process to establish seismology as a science and to investigate its relation to subjects like realism and empiricism in natural sciences, to the nature of explanations and predictions, and to experimental verification and refutation.

Product Details

ISBN-13:	9789814644808
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/22/2016
Pages:	404
Product dimensions:	6.20(w) x 9.00(h) x 1.00(d)

Table of Contents

Foreword vii

List of Figures xix

List of Tables xxi

Acknowledgments xxiii

1 Science of seismology 1

Preliminary remarks 1

1.1 Purpose and methodology: Historical sketch 2

1.2 Classification 10

Closing remarks 11

1.3 Exercises 14

2 Seismology and continuum mechanics 19

Preliminary remarks 19

2.1 On axiomatic formulation 21

2.2 Kinematic descriptions 23

2.2.1 Spacetime 23

2.2.2 Motion 24

2.2.3 Coordinates 27

2.3 Field equations 28

2.3.1 Balance equations 28

2.3.2 Continuity equation 29

2.3.3 Cauchy equation of motion 31

Closing remarks 33

2.4 Exercises 34

3 Hookean solid: Material symmetry 41

Preliminary remarks 41

3.1 Hookean solids 42

3.2 Material symmetry 45

3.2.1 On symmetries 45

3.2.2 On tensor rotations 48

3.2.3 Finite and infinitesimal elasticities 49

3.2.3.1 Deformation gradient 49

3.2.3.2 Elasticity tensor 52

3.2.3.3 Prestressed linearly clastic materials 54

3.2.3.4 Material symmetry: Finite elasticity 57

3.2.3.5 Material symmetry: Relation between finite and infinitesimal elasticities 58

3.2.4 Symmetry classes 61

3.2.4.1 Material-symmetry conditions 61

3.2.4.2 Hooke's law in R³ and R⁶ 64

3.2.4.3 Index symmetries 65

3.2.4.4 Kelvin notation 67

3.2.4.5 Monoclinic tensor 69

3.2.4.6 Orthotropic tensor 75

3.2.4.7 Tetragonal tensor 76

3.2.4.8 Transversely isotropic tensor 78

3.2.4.9 Trigonal tensor 79

3.2.4.10 Cubic tensor 79

3.2.4.11 Isotropic tensor 79

3.2.4.12 Relations among elasticity parameters 80

3.2.4.13 Diclinic solids 85

3.2.4.14 Hexagonal solids 87

Closing remarks 88

3.3 Exercises 89

4 Hookean solid: Effective symmetry and equivalent medium 95

Preliminary remarks 95

4.1 Effective symmetries 96

4.1.1 On accuracy 96

4.1.2 Fixed orientation of coordinate system 103

4.1.2.1 Monoclinic tensor 104

4.1.2.2 Orthotropic tensor 105

4.1.2.3 Tetragonal tensor 106

4.1.2.4 Transversely isotropic tensor 107

4.1.2.5 Trigonal tensor 107

4.1.2.6 Cubic tensor 108

4.1.2.7 Isotropic tensor 109

4.1.3 Optimal orientation of coordinate system 110

4.2 Equivalent media 113

4.2.1 Introduction 113

4.2.2 Equivalence parameters for isotropic layers 116

4.2.2.1 Formulae 116

4.2.2.2 Justification 118

4.2.2.3 Interpretation 129

4.2.3 Equivalence parameters for TI layers 130

Closing remarks 132

4.3 Exercises 134

5 Body waves 159

Preliminary remarks 159

5.1 Wave equations 160

5.1.1 Assumptions and formulation 160

5.1.2 Particular case: Isotropy and homogeneity 161

5.1.3 Particular case: Inhomogeneous string 167

5.1.4 Particular case: String with friction 171

5.2 Solutions of wave equation 171

5.2.1 Introduction 171

5.2.2 Product solution 172

5.2.3 d'Alembert solution 173

5.2.3.1 d'Alembert's approach 173

5.2.3.2 Euler's approach 174

5.2.3.3 Spherical-symmetry approach 177

5.2.4 Fourier-transform solution 178

5.2.5 Green's-function solution 182

5.3 On approximations 185

Closing remarks 188

5.4 Exercises 189

6 Surface, guided and interface waves 197

Preliminary remarks 197

6.1 Introduction 198

6.2 Surface waves: Homogeneous elastic halfspace 200

6.3 Guided waves: Homogeneous layer above halfspace 209

6.3.1 Elastic layer above rigid halfspace 209

6.3.2 Elastic layer above elastic halfspace 212

6.4 Existence of surface and guided waves 216

6.4.1 Introduction 216

6.4.2 Elasticity parameters and mass densities 216

6.4.3 On Love waves in homogeneous halfspace 217

6.4.4 On P waves in homogeneous halfspace 217

6.5 Interface waves: Homogenous halfspaces 219

6.5.1 Introduction 219

6.5.2 Elastic and liquid halfspaces 220

6.5.3 Liquid halfspaces 231

6.6 Existence of interface waves 235

6.6.1 Introduction 235

6.6.2 Elasticity parameters and mass densities 236

6.6.3 On SH waves as interface waves 236

Closing remarks 238

6.7 Exercises 239

7 Variational principles in seismology 241

Preliminary remarks 241

7.1 Historical comments 242

7.2 Fermat's principle 243

7.2.1 Isotropic layered medium 243

7.2.2 Isotropic continuously inhomogeneous medium 246

7.2.3 Global optimization and causality 249

7.2.4 Stationarity versus minimization 251

7.2.5 Mathematical justification 252

7.2.5.1 Fermat's principle 252

7.2.5.2 Head waves 254

7.2.6 Physical interpretation 257

7.2.6.1 Macroscopic interpretation 257

7.2.6.2 Microscopic interpretation 259

7.2.6.3 Phase consideration 259

7.2.7 On teleology of Fermat's principle 260

7.3 Hamilton's principle 264

7.3.1 Action 264

7.3.2 Wave equation 265

7.3.3 Mathematical justification 266

7.3.4 Physical interpretation 267

7.4 Conserved quantities 268

7.4.1 Introduction 268

7.4.2 Ray parameter 268

7.4.2.1 Isotropy 268

7.4.2.2 Anisotropy 270

7.4.3 Hamiltonian and Lagrangian 272

7.4.3.1 Ray theory 272

7.4.3.2 Classical mechanics 274

Closing remarks 275

7.5 Exercises 276

Gravitational and thermal effects in seismology 283

Preliminary remarks 283

8.1 Gravitation 284

5.1.1 Body forces 284

8.1.1 Wave speeds 287

8.2 On weak gravitational waves 291

8.3 Temperature 298

8.3.1 Propagation and diffusion 298

8.3.2 Isothermal and adiabatic formulations 299

8.3.2.1 Lamé parameters 299

8.3.2.2 Bulk moduli 301

Closing remarks 301

8.4 Exercises 303

9 Seismology as science 307

Preliminary remarks 307

9.1 Hypotheticodeductive formulation 308

9.1.1 Hypotheses 308

9.1.2 Deductive argumentation 310

9.2 Theory versus data 313

9.2.1 Introduction 313

9.2.2 Theory-ladenness of data 313

9.2.3 Under determination of theory by data 314

9.3 Bayesian inference 315

9.4 Predictions versus explanations 318

9.4.1 Introduction 318

9.4.2 Covering-Law model 319

9.1.1 Inference to befit explanation 321

9.5 Realistic approach versus instrumental approach 321

9.6 Coherence theory of justification 323

Closing remarks 324

9.7 Exercises 326

Appendix A On covariant and contravariant transformations 331

Preliminary remarks 331

A.1 Contravariant transformations 332

A.2 Covariant transformations 333

A.3 Mixed transformations 334

A.4 Transformations in Cartesian coordinates 334

Closing remarks 335

Appendix B On covariant derivatives 337

Preliminary remarks 337

B.1 Metric tensor 338

B.2 Christoffel symbol 341

B.3 Covariant derivative 342

Closing remarks 344

Appendix C List of symbols 347

C.1 Mathematical relations and operations 347

C.2 Physical quantities 348

C.2.1 Greek letters 348

C.2.2 Roman letters 348

Bibliography 351

Index 363

About the Author 380