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 FunctionsPart 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 MediaPart 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 EquationsPart 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 TheoryPart 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 WavesReferencesAuthor IndexSubject Index