Mathematical Aspects of Seismology (eBook)

Front Cover 1
Mathematical Aspects of Seismology 4
Copyright Page 5
Table of Contents 8
PREFACE 6
Chapter 1. INTRODUCTION 14
1.1 DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS 14
1.2 COORDINATE TRANSFORMATIONS 15
1.3 THE GAMMA AND BETA FUNCTIONS 23
PART I: Integration methods 28
Chapter 2. CONTOUR INTEGRATION AND CONFORMAL TRANSFORMATION 30
2.1 CONTOUR INTEGRATION IN THE COMPLEX PLANE 30
2.2 CONFORMAL TRANSFORMATION 46
Chapter 3. METHODS OF STATIONARY PHASE AND OF STEEPEST DESCENT 56
3.1 METHOD OF STATIONARY PHASE (OR PRINCIPLE OF STATIONARY PHASE) 56
3.2 METHOD OF STEEPEST DESCENT 63
3.3 THE AIRY INTEGRAL 71
Chapter 4. SERIES INTEGRATION 80
4.1 FUNDAMENTAL CONCEPTS 80
4.2 LEGENDRE'S DIFFERENTIAL EQUATION 85
4.3 BESSEL'S DIFFERENTIAL EQUATION 93
4.4 HERMITE'S DIFFERENTIAL EQUATION 100
4.5 LAGUERRE'S DIFFERENTIAL EQUATION 104
4.6 GAUSS' (HYPERGEOMETRIC) DIFFERENTIAL EQUATION—WHITTAKER'S FUNCTIONS 107
4.7 LOVE WAVES IN HETEROGENEOUS ISOTROPIC MEDIA 119
PART II: Special functions 124
Chapter 5. BESSEL FUNCTIONS 126
5.1 ORIGIN OF BESSEL FUNCTIONS 126
5.2. PROPERTIES OF BESSEL COEFFICIENTS 128
5.3 RELATED BESSEL FUNCTIONS 143
5.4 APPLICATIONS OF BESSEL AND HANKEL FUNCTIONS 157
Chapter 6. LEGENDRE FUNCTIONS 168
6.1 LEGENDRE POLYNOMIALS 168
6.2 LEGENDRE FUNCTIONS 177
6.3 APPLICATIONS OF LEGENDRE FUNCTIONS 183
Chapter 7. THE WAVE EQUATION 188
7.1 GENERAL CONSIDERATIONS OF THE WAVE EQUATION 188
7.2 SOLUTIONS OF THE SPACE FORM OF THE WAVE EQUATION 191
7.3 EXPANSION OF A SPHERICAL WAVE INTO PLANE WAVES: SOMMERFELD'S INTEGRAL 200
7.4 KIRCHHOFF'S SOLUTION OF THE WAVE EQUATION 205
7.5 COMMON FEATURES OF SPECIAL FUNCTIONS AND OF SPECIAL DIFFERENTIAL EQUATIONS 214
PART III: Selected mathematical methods 222
Chapter 8. INTEGRAL TRANSFORMS 224
8.1 INTRODUCTION TO LAPLACE AND FOURIER TRANSFORMS 224
8.2 USE OF THE LAPLACE TRANSFORM FOR THE SOLUTION OF DIFFERENTIAL EQUATIONS 240
8.3 IMPULSIVE FUNCTIONS 259
8.4 CAGNIARD'S METHOD 276
Chapter 9. MATRIX CALCULUS 292
9.1 INTRODUCTION 292
9.2 HASKELL'S MATRIX METHOD FOR RAYLEIGH WAVES 293
9.3 LOVE WAVES 308
9.4 BODY-WAVE PROPAGATION THROUGH A MANY-LAYERED MEDIUM 310
Chapter 10. CALCULUS OF VARIATIONS 312
10.1 FUNDAMENTALS OF THE CALCULUS OF VARIATIONS 312
10.2 APPLICATIONS OF THE CALCULUS OF VARIATIONS 319
Chapter 11. INTEGRAL EQUATIONS 328
11.1 DEFINITIONS AND SOLUTIONS OF INTEGRAL EQUATIONS 328
11.2 APPLICATION TO SEISMIC RAY THEORY 334
PART IV: Selected seismological applications 340
Chapter 12. LAMB'S PROBLEM 342
12.1 TWO-DIMENSIONAL PROBLEM IN AN ISOTROPIC ELASTIC SOLID (AREA SOURCE, LINE SOURCE) 342
12.2 THREE-DIMENSIONAL PROBLEM IN AN ISOTROPIC ELASTIC SOLID (VOLUME SOURCE, POINT SOURCE) 365
12.3 ARBITRARY TIME VARIATION IN THE THREE-DIMENSIONAL CASE 375
Chapter 13. WAVE PROPAGATION IN LIQUID MEDIA 380
13.1 WAVE PROPAGATION IN A TWO-LAYERED LIQUID HALF-SPACE 380
13.2 WAVE PROPAGATION IN A LIQUID HALF-SPACE WITH VELOCITY VARYING WITH DEPTH 392
Chapter 14. INFLUENCE OF GRAVITY ON WAVE PROPAGATION 398
14.1 MATHEMATICAL INTRODUCTION 398
14.2 BODY WAVES 402
14.3 SURFACE WAVES 413
REFERENCES 414
AUTHOR INDEX 418
SUBJECT INDEX 422