Waves and Rays in Elastic Continua

Description

Seismology, as a branch of mathematical physics, is an active subject of both research and development. Its reliance on computational and technological advances continuously motivates the developments of its underlying theory. The fourth edition of Waves and Rays in Elastic Continua responds to these needs. The book is both a research reference and a textbook. Its careful and explanatory style, which includes numerous exercises with detailed solutions, makes it an excellent textbook for the senior undergraduate and graduate courses, as well as for an independent study. Used in its entirety, the book could serve as a sole textbook for a year-long course in quantitative seismology. Its parts, however, are designed to be used independently for shorter courses with different emphases. The book is not limited to quantitive seismology; it can serve as a textbook for courses in mathematical physics or applied mathematics. Contents Dedication Acknowledgments Preface to First Edition Changes from First Edition Changes from Second Edition Changes from Third Edition List of Figures Part 1. Elastic continua Introduction to Part 1 Chapter 1. Deformations Preliminary Remarks 1.1. Notion of Continuum 1.2. Rudiments of Continuum Mechanics 1.2.1. Axiomatic format 1.2.2. Primitive concepts of continuum mechanics 1.3. Material and Spatial Descriptions 1.3.1. Fundamental concepts 1.3.2. Material time derivative 1.3.3. Conditions of linearized theory 1.4. Strain 1.4.1. Introductory comments 1.4.2. Derivation of strain tensor 1.4.3. Physical meaning of strain tensor 1.5. Rotation Tensor and Rotation Vector Closing Remarks 1.6. Exercises Chapter 2. Forces and Balance Principles Preliminary Remarks 2.1. Conservation of Mass 2.1.1. Introductory comments 2.1.2. Integral equation 2.1.3. Equation of continuity 2.2. Time Derivative of Volume Integral 2.3. Stress 2.3.1. Stress as description of surface forces 2.3.2. Traction 2.4. Balance of Linear Momentum 2.5. Stress Tensor 2.5.1. Traction on coordinate planes 2.5.2. Traction on arbitrary planes 2.6. Cauchy's Equations of Motion 2.6.1. General formulation 2.6.2. Surface-forces formulation 2.7. Balance of Angular Momentum 2.7.1. Introductory comments 2.7.2. Integral equation 2.7.3. Symmetry of stress tensor 2.8. Fundamental Equations Closing Remarks 2.9. Exercises Chapter 3. Stress-Strain Equations Preliminary Remarks 3.1. Rudiments of Constitutive Equations 3.2. Formulation of Stress-Strain Equations: Hookean Solid 3.2.1. Introductory comments 3.2.2. Tensor form 3.2.3. Matrix form 3.3. Determined System 3.4. Anelasticity 3.4.1. Introductory comments 3.4.2. Viscosity: Stokesian fluid 3.4.3. Viscoelasticity: Kelvin-Voigt model 3.4.4. Viscoelasticity: Maxwell model Closing Remarks 3.5. Exercises Chapter 4. Strain Energy Preliminary remarks 4.1. Strain-energy Function 4.2. Strain-energy Function and Elasticity-tensor Symmetry 4.2.1. Fundamental considerations 4.2.2. Elasticity parameters 4.2.3. Matrix form of stress-strain equations 4.2.4. Coordinate transformations 4.3. Stability Conditions 4.3.1. Physical justification 4.3.2. Mathematical formulation 4.3.3. Constraints on elasticity parameters 4.4. System of Equations for Elastic Continua 4.4.1. Elastic continua 4.4.2. Governing equations Closing Remarks 4.5. Exercises Chapter 5. Material Symmetry Preliminary remarks 5.1. Orthogonal Transformations 5.1.1. Transformation matrix 5.1.2. Symmetry group 5.2. Transformation of Coordinates 5.2.1. Introductory comments 5.2.2. Transformation of stress-tensor components 5.2.3. Transformation of strain-tensor components 5.2.4. Stress-strain equations in transformed coordinates 5.2.5. On matrix forms 5.3. Condition for Material Symmetry 5.4. Point Symmetry 5.5. Generally Anisotropic Continuum 5.6. Monoclinic Continuum 5.6.1. Elasticity matrix 5.6.2. Vanishing of tensor components 5.6.3. Natural coordinate system 5.7. Orthotropic Continuum 5.8. Trigonal Continuum 5.8.1. Elasticity matrix 5.8.2. Natural coordinate system 5.9. Tetragonal Continuum 5.9.1. Elasticity matrix 5.9.2. Natural coordinate system 5.10. Transversely Isotropic Continuum 5.10.1. Elasticity matrix 5.10.2. Rotation invariance 5.11. Cubic Continuum 5.12. Isotropic Continuum 5.12.1. Elasticity matrix 5.12.2. Lamé's parameters 5.12.3. Tensor formulation 5.12.4. Physical meaning of Lamé's parameters 5.13. Relations Among Symmetry Classes Closing Remarks 5.14. Exercises Part 2. Waves and rays Introduction to Part 2 Chapter 6. Equations of Motion: Isotropic Homogeneous Continua Preliminary Remarks 6.1. Wave Equations 6.1.1. Equation of motion 6.1.2. Wave equation for P waves 6.1.3. Wave equation for S waves 6.1.4. Physical interpretation 6.2. Plane Waves 6.3. Displacement Potentials 6.3.1. Helmholtz's decomposition 6.3.2. Gauge transformation 6.3.3. Equation of motion 6.3.4. P and S waves 6.4. P and S Waves in Terms of Displacements 6.4.1. Introductory comments 6.4.2. Sufficient conditions for P and S waves 6.4.3. Necessary conditions for P and S waves 6.5. Solutions of Wave Equation for Single Spatial Dimension 6.5.1. d'Alembert's approach 6.5.2. Directional derivative 6.5.3. Well-posed problem 6.5.4. Causality, finite propagation speed and sharpness of signals 6.6. Solution of Wave Equation for Two and Three Spatial Dimensions 6.6.1. Introductory comments 6.6.2. Three spatial dimensions 6.6.3. Two spatial dimensions 6.7. On Evolution Equation 6.8. Solutions of Wave Equation for One-Dimensional Scattering 6.9. On Weak Solutions of Wave Equation 6.9.1. Introductory comments 6.9.2. Weak derivatives 6.9.3. Weak solution of wave equation 6.10. Reduced Wave Equation 6.10.1. Harmonic-wave trial solution 6.10.2. Fourier's transform of wave equation 6.11. Extensions of Wave Equation 6.11.1. Introductory comments 6.11.2. Standard wave equation 6.11.3. Wave equation and elliptical velocity dependence 6.11.4. Wave equation and weak inhomogeneity Closing Remarks 6.12. Exercises Chapter 7. Equations of Motion: Aniso-tropic Inhomogeneous Continua Preliminary Remarks 7.1. Formulation of Equations 7.2. Formulation of Solutions 7.2.1. Introductory comments 7.2.2. Trial-solution formulation: General wave 7.2.3. Trial-solution formulation: Harmonic wave 7.2.4. Asymptotic-series formulation 7.3. Eikonal Equation Closing Remarks 7.4. Exercises Chapter 8. Hamilton's Ray Equations Preliminary Remarks 8.1. Method of Characteristics 8.1.1. Level-set functions 8.1.2. Characteristic equations 8.1.3. Consistency of formulation 8.2. Time Parametrization of Characteristic Equations 8.2.1. General formulation 8.2.2. Equations with variable scaling factor 8.2.3. Equations with constant scaling factor 8.2.4. Formulation of Hamilton's ray equations 8.3. Physical Interpretation of Hamilton's Ray Equations and Solutions 8.3.1. Equations 8.3.2. Solutions 8.4. Relation between p and 8.4.1. General formulation 8.4.2. Phase and ray velocities 8.4.3. Phase and ray angles 8.4.4. Geometrical illustration 8.5. Example: Elliptical Anisotropy and Linear Inhomogeneity 8.5.1. Introductory comments 8.5.2. Eikonal equation 8.5.3. Hamilton's ray equations 8.5.4. Initial conditions 8.5.5. Physical interpretation of equations and conditions 8.5.6. Solution of Hamilton's ray equations 8.5.7. Solution of eikonal equation 8.5.8. Physical interpretation of solutions 8.6. Example: Isotropy and Inhomogeneity 8.6.1. Parametric form 8.6.2. Explicit form Closing Remarks 8.7. Exercises Chapter 9. Christoffel's Equations Preliminary Remarks 9.1. Explicit form of Christoffel's Equations 9.2. Christoffel's Equations and Anisotropic Continua 9.2.1. Introductory comments 9.2.2. Monoclinic continua 9.2.3. Transversely isotropic continua 9.3. Phase-slowness Surfaces 9.3.1. Introductory comments 9.3.2. Convexity of innermost sheet 9.3.3. Intersection points Closing Remarks 9.4. Exercises Chapter 10. Reflection and Transmission Preliminary Remarks 10.1. Angles at Interface 10.1.1. Phase angles 10.1.2. Ray angles 10.1.3. Example: Elliptical velocity dependence 10.2. Amplitudes at Interface 10.2.1. Kinematic and dynamic boundary conditions 10.2.2. Reflection and transmission amplitudes Closing Remarks 10.3. Exercises Chapter 11. Lagrange's Ray Equations Preliminary Remarks 11.1. Legendre's Transformation of Hamiltonian 11.2. Formulation of Lagrange's Ray Equations 11.3. Beltrami's Identity Closing Remarks 11.4. Exercises Part 3. Variational formulation of rays Introduction to Part 3 Chapter 12. Euler's Equations Preliminary Remarks 12.1. Mathematical Background 12.2. Formulation of Euler's Equation 12.3. Beltrami's Identity 12.4. Generalizations of Euler's Equation 12.4.1. Introductory comments 12.4.2. Case of several variables 12.4.3. Case of several functions 12.4.4. Higher-order derivatives 12.5. Special Cases of Euler's Equation 12.5.1. Introductory comments 12.5.2. Independence of z 12.5.3. Independence of x and z 12.5.4. Independence of x 12.5.5. Total derivative 12.5.6. Function of x and z 12.6. First Integrals 12.7. Lagrange's Ray Equations as Euler's Equations Closing Remarks 12.8. Exercises Chapter 13. Variational Principles Preliminary Remarks 13.1. Fermat's Principle 13.1.1. Statement of Fermat's principle 13.1.2. Properties of Hamiltonian H 13.1.3. Variational equivalent of Hamilton's ray equations 13.1.4. Properties of Lagrangian L 13.1.5. Parameter-independent Lagrange's ray equations 13.1.6. Ray velocity 13.1.7. Proof of Fermat's principle 13.2. Hamilton's Principle: Example 13.2.1. Introductory comments 13.2.2. Action 13.2.3. Lagrange's equations of motion 13.2.4. Wave equation Closing Remarks 13.3. Exercises Chapter 14. Ray Parameters Preliminary Remarks 14.1. Traveltime Integrals 14.2. Ray Parameters as First Integrals 14.3. Example: Elliptical Anisotropy and Linear Inhomogeneity 14.3.1. Introductory comments 14.3.2. Rays 14.3.3. Traveltimes 14.4. Rays in Isotropic Continua 14.5. Lagrange's Ray Equations in xz-Plane 14.6. Conserved Quantities and Hamilton's Ray Equations Closing Remarks 14.7. Exercises Part 4. Appendices Introduction to Part 4 Appendix A. Euler's Homogeneous-Function Theorem Preliminary Remarks A.1. Homogeneous Functions A.2. Homogeneous-Function Theorem Closing Remarks Appendix B. Legendre's Transformation Preliminary Remarks B.1. Geometrical Context B.1.1. Surface and its tangent planes B.1.2. Single-variable case B.2. Duality of Transformation B.3. Transformation between Lagrangian L and Hamiltonian H B.4. Transformation and Ray Equations Closing Remarks Appendix C. List of Symbols C.1. Mathematical Relations and Operations C.2. Physical Quantities C.2.1. Greek letters C.2.2. Roman letters Bibliography Index About the Author