Mathematical Aspects of Seismology

Mathematical Aspects of Seismology

Developments in Solid Earth Geophysics

1st Edition - January 1, 1968

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  • Author: Markus Båth
  • eBook ISBN: 9781483274973

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Description

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.

Table of Contents


  • 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

Product details

  • No. of pages: 428
  • Language: English
  • Copyright: © Elsevier 1968
  • Published: January 1, 1968
  • Imprint: Elsevier
  • eBook ISBN: 9781483274973

About the Author

Markus Båth

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