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E-raamat: Seismic Waves and Rays in Elastic Media

M.A. Slawinski (Department of Earth Sciences, Memorial University of Newfoundland, St. John's, NF, Canada)

Formaat: PDF+DRM
Sari: Handbook of Geophysical Exploration: Seismic Exploration
Ilmumisaeg: 04-Aug-2003
Kirjastus: Elsevier Science Ltd
Keel: eng
ISBN-13: 9780080540894

Teised raamatud teemal:

Analytical mechanics

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Formaat: PDF+DRM
Sari: Handbook of Geophysical Exploration: Seismic Exploration
Ilmumisaeg: 04-Aug-2003
Kirjastus: Elsevier Science Ltd
Keel: eng
ISBN-13: 9780080540894

Teised raamatud teemal:

Analytical mechanics

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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.

Arvustused

"The purpose of this title...is to use aspects of continuum mechanics, wave theory, and ray theory to describe phenomena resulting from the propagation of waves." --M.A. Slawinski, FIRST BREAK, Volume 23

I Elastic continua

(124)

Introduction to Part I

(4)

Deformations

(26)

Preliminary remarks

(1)

Notion of continuum

(1)

Material and spatial descriptions

(6)

Fundamental concepts

(1)

Material time derivative

(2)

Conditions of linearized theory

(3)

Strain

(9)

Derivation of strain tensor

(4)

Physical meaning of strain tensor

(5)

Rotation tensor and rotation vector

(9)

Closing remarks

(1)

Exercises

(8)

Forces and balance principles

(28)

Preliminary remarks

(1)

Conservation of mass

(2)

Integral equation

(2)

Equation of continuity

(1)

Time derivative of volume integral

(2)

Stress

(1)

Stress as description of surface forces

(1)

Traction

(1)

Balance of linear momentum

(1)

Stress tensor

(7)

Traction on coordinate planes

(3)

Traction on arbitrary planes

(4)

Cauchy's equations of motion

(4)

General formulation

(2)

Example: Surface-forces formulation

(2)

Balance of angular momentum

(3)

Integral equation

(1)

Symmetry of stress tensor

(2)

Fundamental equations

(7)

Closing remarks

(1)

Exercises

(5)

Stress-strain equations

(8)

Preliminary remarks

(1)

Formulation of stress-strain equations

(4)

Tensorial form

(1)

Matrix form

(2)

Determined system

(3)

Closing remarks

(1)

Exercises

(1)

Strain energy

(16)

Preliminary remarks

(1)

Strain-energy function

(1)

Strain-energy function and elasticity-tensor symmetry

(3)

Fundamental considerations

(2)

Stress-strain equations

(1)

Coordinate transformations

(1)

Stability conditions

(2)

Physical justification

(1)

Mathematical formulation

(1)

Constraints on elasticity parameters

(1)

System of equations for elastic continua

(9)

Elastic continua

(1)

Governing equations

(2)

Closing remarks

(1)

Exercises

(5)

Material symmetry

(40)

Preliminary remarks

(1)

Orthogonal transformations

(1)

Transformation matrix

(1)

Symmetry group

(1)

Transformation of coordinates

(5)

Transformation of stress-tensor components

(3)

Transformation of strain-tensor components

(1)

Stress-strain equations in transformed coordinates

(1)

Condition for material symmetry

(2)

Point symmetry

(1)

Generally anisotropic continuum

(1)

Monoclinic continuum

(3)

Elasticity matrix

(1)

Natural coordinate system

(2)

Orthotropic continuum

(2)

Tetragonal continuum

100

(1)

Transversely isotropic continuum

101

(5)

Elasticity matrix

101

(1)

Rotation invariance

102

(4)

Isotropic continuum

106

(19)

Elasticity matrix

106

(1)

Lame's parameters

107

(1)

Tensorial formulation

108

(1)

Physical meaning of Lame's parameters

109

(1)

Closing remarks

110

(1)

Exercises

111

(14)

II Waves and rays

125

(140)

Introduction to Part II

127

(2)

Equations of motion: Isotropic homogeneous continua

129

(34)

Preliminary remarks

129

(1)

Wave equations

130

(4)

Equation of motion

130

(2)

Wave equation for P waves

132

(1)

Wave equation for S waves

133

(1)

Plane waves

134

(3)

Displacement potentials

137

(4)

Helmholtz's decomposition

137

(1)

Equation of motion

138

(1)

P and S waves

139

(2)

Solutions of one-dimensional wave equation

141

(3)

Reduced wave equation

144

(1)

Extensions of wave equation

145

(18)

Standard wave equation

146

(1)

Wave equation and elliptical velocity dependence

146

(3)

Wave equation and weak inhomogeneity

149

(5)

Closing remarks

154

(1)

Exercises

155

(8)

Equations of motion: Anisotropic inhomogeneous continua

163

(10)

Preliminary remarks

163

(1)

Formulation of equations

164

(1)

Formulation of solutions

165

(2)

Eikonal equation

167

(6)

Closing remarks

170

(1)

Exercises

170

(3)

Hamilton's ray equations

173

(24)

Preliminary remarks

173

(1)

Method of characteristics

174

(5)

Level-set functions

174

(1)

Characteristic equations

175

(3)

Consistency of formulation

178

(1)

Time parametrization of characteristic equations

179

(4)

General formulation

179

(1)

Equations with variable scaling factor

180

(1)

Equations with constant scaling factor

181

(1)

Formulation of Hamilton's ray equations

182

(1)

Example: Ray equations in isotropic inhomogeneous continua

183

(14)

Parametric form

183

(1)

Explicit form

184

(1)

Closing remarks

185

(1)

Exercises

186

(11)

Lagrange's ray equations

197

(20)

Preliminary remarks

197

(1)

Transformation of Hamilton's ray equations

198

(3)

Formulation of Lagrange's ray equations

198

(2)

Beltrami's identity

200

(1)

Relation between p and x

201

(16)

Phase and ray velocities

201

(3)

Phase and ray angles

204

(2)

Geometrical illustration

206

(1)

Closing remarks

206

(1)

Exercises

207

(10)

Christoffel's equations

217

(28)

Preliminary remarks

217

(1)

Explicit form of Christoffel's equations

218

(3)

Christoffel's equations and anisotropic continua

221

(11)

Monoclinic continua

222

(4)

Transversely isotropic continua

226

(6)

Phase-slowness surfaces

232

(13)

Convexity of innermost sheet

233

(1)

Intersection points

233

(2)

Closing remarks

235

(1)

Exercises

236

(9)

Reflection and transmission

245

(20)

Preliminary remarks

245

(1)

Angles at interface

246

(5)

Phase angles

246

(1)

Ray angles

247

(1)

Example: Elliptical velocity dependence

248

(3)

Amplitudes at interface

251

(14)

Kinematic and dynamic boundary conditions

251

(4)

Reflection and transmission amplitudes

255

(4)

Closing remarks

259

(1)

Exercises

260

(5)

III Variational formulation of rays

265

(72)

Introduction to Part III

267

(2)

Euler's equations

269

(24)

Preliminary remarks

269

(1)

Mathematical background

270

(1)

Formulation of Euler's equation

271

(3)

Beltrami's identity

274

(1)

Generalizations of Euler's equation

274

(3)

Case of several variables

274

(1)

Case of several functions

275

(1)

Higher-order derivatives

276

(1)

Special cases of Euler's equation

277

(5)

Independence of z

277

(1)

Independence of x and z

277

(1)

Independence of x

278

(1)

Total derivative

279

(1)

Function of x and z

279

(3)

First integrals

282

(1)

Lagrange's ray equations as Euler's equations

283

(10)

Closing remarks

284

(1)

Exercises

284

(9)

Fermat's principle

293

(26)

Preliminary remarks

293

(1)

Formulation of Fermat's principle

294

(7)

Statement of Fermat's principle

294

(1)

Properties of Hamiltonian H

294

(2)

Variational equivalent of Hamilton's ray equations

296

(1)

Properties of Lagrangian L

296

(2)

Parameter-independent Lagrange's ray equations

298

(1)

Ray velocity

299

(1)

Proof of Fermat's principle

300

(1)

Illustration of Hamilton's principle

301

(18)

Action

301

(2)

Lagrange's equations of motion

303

(1)

Wave equation

304

(4)

Closing remarks

308

(1)

Exercises

308

(11)

Ray parameters

319

(18)

Preliminary remarks

319

(1)

Traveltime integrals

320

(1)

Ray parameters as first integrals

320

(2)

Example: Ellipticity and linearity

322

(6)

Rays

323

(3)

Traveltimes

326

(1)

Isotropic extension

327

(1)

Rays in isotropic continua

328

(1)

Lagrange's ray equations in xz-plane

329

(2)

Conserved quantities and Hamilton's ray equations

331

(6)

Closing remarks

332

(1)

Exercises

332

(5)

IV Appendices

337

(4)

Introduction to Part IV

339

(2)

A Euler's homogeneous-function theorem

341

(6)

Preliminary remarks

341

(1)

A.1 Homogeneous functions

342

(1)

A.2 Homogeneous-function theorem

343

(4)

Closing remarks

345

(2)

B Legendre's transformation

347

(8)

Preliminary remarks

347

(1)

B.1 Geometrical context

348

(2)

B.1.1 Surface and its tangent planes

348

(1)

B.1.2 Single-variable case

348

(2)

B.2 Duality of transformation

350

(1)

B.3 Transformation between L and H

350

(2)

B.4 Transformation and ray equations

352

(3)

Closing remarks

353

(2)

C List of symbols

355

(4)

C.1 Mathematical relations and operations

355

(2)

C.2 Physical quantities

357

(2)

C.2.1 Greek letters

357

(1)

C.2.2 Roman letters

358

(1)

Bibliography

359

(15)

Index

374

(27)

About the author

401

http://www.rug.nl/rechten/faculteit/vakgroepen/beof/ecof/mdw/WoerdmanE

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