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.
Author(s): Michael A. Slawinski
Edition: 4
Publisher: World Scientific Publishing
Year: 2020
Language: English
Pages: 679
City: New Jersey
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
