Seismic Waves and Rays in Elastic Media (eBook)

M.A. Slawinski (Autor)

eBook Download: PDF

2003 | 1. Auflage
424 Seiten
Elsevier Science (Verlag)
978-0-08-054089-4 (ISBN)

Infos

This book seeks to explore seismic phenomena in elastic media and emphasizes the interdependence of mathematical formulation and physical meaning. The purpose of this title - which is intended for senior undergraduate and graduate students as well as scientists interested in quantitative seismology - is to use aspects of continuum mechanics, wave theory and ray theory to describe phenomena resulting from the propagation of waves.

The book is divided into three parts: Elastic continua, Waves and rays, and Variational formulation of rays. In Part I, continuum mechanics are used to describe the material through which seismic waves propagate, and to formulate a system of equations to study the behaviour of such material. In Part II, these equations are used to identify the types of body waves propagating in elastic continua as well as to express their velocities and displacements in terms of the properties of these continua. To solve the equations of motion in anisotropic inhomogeneous continua, the high-frequency approximation is used and establishes the concept of a ray. In Part III, it is shown that in elastic continua a ray is tantamount to a trajectory along which a seismic signal propagates in accordance with the variational principle of stationary travel time.

Inhalt

Front Cover 1
Seismic Waves and Rays in Elastic Media 4
Copyright Page 5
Contents 16
Part I: Elastic continua 24
Introduction to Part I 26
Chapter 1. Deformations 30
1.1 Notion of continuum 31
1.2 Material and spatial descriptions 32
1.3 Strain 38
1.4 Rotation tensor and rotation vector 47
Chapter 2. Forces and balance principles 56
2.1 Conservation of mass 57
2.2 Time derivative of volume integral 59
2.3 Stress 61
2.4 Balance of linear momentum 62
2.5 Stress tensor 63
2.6 Cauchy's equations of motion 70
2.7 Balance of angular momentum 74
2.8 Fundamental equations 77
Chapter 3. Stress - strain equations 84
3.1 Formulation of stress-strain equations 85
3.2 Determined system 89
Chapter 4. Strain energy 92
4.1 Strain-energy function 93
4.2 Strain-energy function and elasticity-tensor symmetry 94
4.3 Stability conditions 97
4.4 System of equations for elastic continua 99
Chapter 5. Material symmetry 108
5.1 Orthogonal transformations 109
5.2 Transformation of coordinates 110
5.3 Condition for material symmetry 115
5.4 Point symmetry 117
5.5 Generally anisotropic continuum 117
5.6 Monoclinic continuum 118
5.7 Orthotropic continuum 121
5.8 Tetragonal continuum 123
5.9 Transversely isotropic continuum 124
5.10 Isotropic continuum 129
Part II: Waves and rays 148
Introduction to Part II 150
Chapter 6. Equations of motion: Isotropic homogeneous continua 152
6.1 Wave equations 153
6.2 Plane waves 157
6.3 Displacement potentials 160
6.4 Solutions of one-dimensional wave equation 164
6.5 Reduced wave equation 167
6.6 Extensions of wave equation 168
Chapter 7. Equations of motion: Anisotropic inhomogeneous continua 186
7.1 Formulation of equations 187
7.2 Formulation of solutions 188
7.3 Eikonal equation 190
Chapter 8. Hamilton' s ray equations 196
8.1 Method of characteristics 197
8.2 Time parametrization of characteristic equations 202
8.3 Example: Ray equations in isotropic inhomogeneous continua 206
Chapter 9. Lagrange's ray equations 220
9.1 Transformation of Hamilton's ray equations 221
9.2 Relation between p and x 224
Chapter 10. Christoffel' s equations 240
10.1 Explicit form of Christoffel's equations 241
10.2 Christoffel's equations and anisotropic continua 244
10.3 Phase-slowness surfaces 255
Chapter 11. Reflection and transmission 268
11.1 Angles at interface 269
11.2 Amplitudes at interface 274
Part III: Variational formulation of rays 288
Introduction to Part III 290
Chapter 12. Euler's equations 292
12.1 Mathematical background 293
12.2 Formulation of Euler's equation 294
12.3 Beltrami's identity 297
12.4 Generalizations of Euler's equation 297
12.5 Special cases of Euler's equation 300
12.6 First integrals 305
12.7 Lagrange's ray equations as Euler's equations 306
Chapter 13. Fermat's principle 316
13.1 Formulation of Fermat ' s principle 317
13.2 Illustration of Hamilton's principle 324
Chapter 14. Ray parameters 342
14.1 Traveltime integrals 343
14.2 Ray parameters as first integrals 343
14.3 Example: Ellipticity and linearity 345
14.4 Rays in isotropic continua 351
14.5 Lagrange's ray equations in xz-plane 352
14.6 Conserved quantities and Hamilton's ray equations 354
Part IV: Appendices 360
Introduction to Part IV 362
Appendix A. Euler' s homogeneous-function theorem 364
A.1 Homogeneous functions 365
A.2 Homogeneous-function theorem 366
Appendix B. Legendre's transformation 370
B.1 Geometrical context 371
B.2 Duality of transformation 373
B.3 Transformation between and L and H 373
B.4 Transformation and ray equations 375
Appendix C. List of symbols 378
C.1 Mathematical relations and operations 378
C.2 Physical quantities 380
Bibliography 382
Index 397
About the author 424

Produkt-Details

• Erscheint lt. Verlag: 4.8.2003
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geologie
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geophysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Technik
• ISBN-10: 0-08-054089-9 / 0080540899
• ISBN-13: 978-0-08-054089-4 / 9780080540894