Mathematical Aspects of Seismology

Developments in Solid Earth Geophysics, 4: Mathematical Aspects of Seismology introduces studies of the more advanced parts of theoretical seismology. The manuscript first ponders on contour integration and conformal transformation, methods of stationary phase and steepest descent, and series integration. Discussions focus on Love waves in heterogeneous isotropic media, Laguerre's differential equation, Hermite's differential equation, method of steepest descent, method of stationary phase, contour integration in the complex plane, and conformal transformation. The text then examines series integration, Bessel functions, Legendre functions, and wave equations. Topics include general considerations of the wave equation, expansion of a spherical wave into plane waves, common features of special functions and special differential equations, applications of Legendre functions, Legendre polynomials, Bessel's differential equation, and properties of Bessel coefficients. The book explores the influence of gravity on wave propagation, matrix calculus, wave propagation in liquid media, integral equations, calculus of variations, and integral transforms. The text is a valuable source of data for researchers wanting to study the mathematical aspects of seismology.

Preface

Chapter 1. Introduction

1.1 Differential Equations of Mathematical Physics

1.2 Coordinate Transformations

1.3 The Gamma and Beta Functions

Part I. Integration Methods

Chapter 2. Contour Integration and Conformal Transformation

2.1 Contour Integration in the Complex Plane

2.2 Conformal Transformation

Chapter 3. Methods of Stationary Phase and of Steepest Descent

3.1 Method of Stationary Phase (or Principle of Stationary Phase)

3.2 Method of Steepest Descent

3.3 The Airy Integral

Chapter 4. Series Integration

4.1 Fundamental Concepts

4.2 Legendre's Differential Equation

4.3 Bessel's Differential Equation

4.4 Hermite's Differential Equation

4.5 Laguerre's Differential Equation

4.6 Gauss' (Hypergeometric) Differential Equation—Whittaker's Functions

4.7 Love Waves in Heterogeneous Isotropic Media

Part II. Special Functions

Chapter 5. Bessel Functions

5.1 Origin of Bessel Functions

5.2 Properties of Bessel Coefficients

5.3 Related Bessel Functions

5.4 Applications of Bessel and Hankel Functions

Chapter 6. Legendre Functions

6.1 Legendre Polynomials

6.2 Legendre Functions

6.3 Applications of Legendre Functions

Chapter 7. The Wave Equation

7.1 General Considerations of the Wave Equation

7.2 Solution of the Space Form of the Wave Equation

7.3 Expansion of a Spherical Wave into Plane Waves: Sommerfeld's Integral

7.4 Kirchhoff's Solution of the Wave Equation

7.5 Common Features of Special Functions and of Special Differential Equations

Part III. Selected Mathematical Methods

Chapter 8. Integral Transforms

8.1 Introduction to Laplace and Fourier Transforms

8.2 Use of the Laplace Transform for the Solution of Differential Equations

8.3 Impulsive Functions

8.4 Cagniard's Method

Chapter 9. Matrix Calculus

9.1 Introduction

9.2 Haskell's Matrix Method for Rayleigh Waves

9.3 Love Waves

9.4 Body-Wave Propagation through a Many-Layered Medium

Chapter 10. Calculus of Variations

10.1 Fundamentals of the Calculus of Variations

10.2 Applications of the Calculus of Variations

Chapter 11. Integral Equations

11.1 Definitions and Solutions of Integral Equations

11.2 Application to Seismic Ray Theory

Part IV. Selected Seismological Applications

Chapter 12. Lamb's Problem

12.1 Two-Dimensional Problem in an Isotropic Elastic Solid (Area Souce, Line Source)

12.2 Three-Dimensional Problem in an Isotropic Elastic Solid (Volume Source, Point Source)

12.3 Aibitrary Time Variation in the Three-Dimensional Case

Chapter 13. Wave Propagation in Liquid Media

13.1 Wave Propagation in a Two-Layered Liquid Half-Space

13.2 Wave Propagation in a Liquid Half-Space with Velocity Varying with Depth

Chapter 14. Influence of Gravity on Wave Propagation

14.1 Mathematical Introduction

14.2 Body Waves

14.3 Surface Waves

References

Author Index

Subject Index