Mechanics of Continua and Wave Dynamics

353 Pages·1985·10.631 MB·English

by  Academician Leonid Brekhovskikh, Dr. Valery Goncharov (auth.)

Springer Series on
1
Springer Series on
V;ave Pbenomena
Editor: Leopold B. Felsen
Volume 1 Mechanics of Continua and Wave Dynamics
By L. Brekhovskikh, V. Goncharov
L. Brekhovskikh V. Goncharov
Mechanics of Continua
and Wave Dynamics
With 99 Figures
Springer-¥erlag
Berlin Heidelberg New York Tokyo
Academician Leonid Brekhovskikh
Dr. Valery Goncharov
P. P. Shirsov Institute of Oceanology, Academy of Sciences of the USSR, Krasikowa 23
SU-1l72l8 Moscow, USSR
Series Editor:
Professor Leopold B. Felsen Ph. D.
Polytechnic Institute of New York, 333 Jay Street, Brooklyn, NY 11201, USA
Title of the original Russian edition: VVedenie v mekhaniku sploshnykh sred
© "Nauka" Publishing House, Moscow 1982
ISBN-13: 978-3-642-96863-1 e-ISBN-13: 978-3-642-96861-7
DOl: 10.1007/978-3-642-96861-7
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction
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Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1985
Softcover reprint of the hardcover 1st edition 1985
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence
of a specific statement, that such names are exempt from the relevant protective laws and regulations
and therefore free for general use.
Typesetting: Syntax International, Singapore 0513
2153/3130-543210
Preface
This text is based on lectures given by the authors to students of the Physico
Technical Institute in Moscow in the course of many years. Teaching mechan
ics of continuous media to students in physics has some specific features. Un
fortunately, the students rarely have enough time for this branch of science.
Over a comparatively short period of time and without sophisticated mathe
matics, the lecturer has to set forth the principal facts and methods of this
rather important aspect of theoretical physics. The goal can be achieved only
if the knowledge and intuition obtained by students in other courses has been
mobilized efficiently. These observations have extensively been taken into
account when classical as well as more contemporary and actively developing
branches of mechanics of continua are considered.
The theory of wave propagation is the most important topic in mechanics
of continua for those who work in the field of physics and geophysics. That is
why most attention is paid to this topic in the main text as well as in numerous
exercises. The propagation of various kinds of hydrodynamic, magnetohydro
dynamic, acoustic and seismic waves is considered in some detail.
We begin with simple questions and proceed to more complicated ones.
Accordingly, we treat the elasticity theory first. In the limits this appears to be
simpler than the mechanics of fluids. For further study of each topic the
reader will find references in the bibliography.
The authors would like to thank very much G. 1. Barenblat, A. G.
Voronovich, G. S. Golitzin and L. A. Ostrovsky for reading the manuscipt
and for many useful comments, and also V. Vavilova for translating the
greater part of the book from Russian into English.
December 1984 L. Brekhovskikh . V. Goncharov
Contents
Part I Theory of Elasticity
1. The Main Types of Strain in Elastic Solids ...................... 2
1.1 Equations of Linear Elasticity Theory ...................... 2
1.1.1 Hooke's Law ..................................... 2
1.1.2 Differential Form of Hooke's Law. Principle of
Superposition ..................................... 3
1.2 Homogeneous Strains ................................... 4
1.2.1 An Elastic Body Under the Action of Hydrostatic Pressure 4
1.2.2 Longitudinal Strain with Lateral Displacements
Forbidden ........................................ 5
1.2.3 PUre Shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Heterogeneous Strains ................................... 8
1.3.1 Torsion of a Rod .................................. 8
1.3.2 Bending of a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.3 Shape of a Beam Under Load. . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Exercises .............................................. 12
2. Waves in Rods, Vibrations of Rods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Longitudinal Waves ..................................... 18
2.1.1 Wave Equation .................................... 18
2.1.2 Harmonic Waves .................................. 19
2.2 Reflection of Longitudinal Waves ......................... 19
2.2.1 Boundary Conditions .............................. 19
2.2.2 Wave Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Longitudinal Oscillations of Rods ......................... 22
2.4 Torsional Waves in a Rod. Torsional Vibrations ............. 23
2.5 Bending Waves in Rods .................................. 24
2.5.1 The Equation for Bending Waves .................... 24
2.5.2 Boundary Conditions. Harmonic Waves .............. 26
2.5.3 Reflection of Waves. Bending Vibrations .............. 27
2.6 Wave Dispersion and Group Velocity ...................... 28
2.6.1 Propagation of Nonharmonic Waves ................. 28
2.6.2 Propagation of Narrow-Band Disturbances. . . . . . . . . . . . 29
2.7 Exercises .............................................. 30
VIII Contents
3. General Theory of Stress and Strain ........................... 39
3.1 Description of the State ofaDeformed Solid ................ 39
3.1.1 Stress Tensor. ... ... . .. . . . . . . . ... . . .. . . .. . . ... . .. . . 39
3.1.2 The Strain Tensor. .. . ..... . . . . . .. . . .. . . .. ... .. . . .. . 41
3.1.3 The Physical Meaning of the Strain Tensor's Components 42
3.2 Equations of Motion for a Continuous Medium . . . . . . . . . . . . . . 44
3.2.1 Derivation of the Equation of Motion . . . . . . . . . . . . . . . . . 44
3.2.2 Strain-Stress Relation. Elasticity Tensor. . . . . . . . . . . . . . . 45
3.3 The Energy of a Deformed Body .......................... 45
3.3.1 The Energy Density ................................ 45
3.3.2 The Number of Independent Components of the
Elasticity Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 The Elastic Behaviour ofIsotropic Bodies. . . . . . . . . . . . . . . . . . . 48
3.4.1 The Generalized Hooke's Law for an Isotropic Body .... 48
3.4.2 The Relationship Between Lame's Constants and E and v 48
3.4.3 The Equations of Motion for an Isotropic Medium. . . . . . 49
3.5 Exercises .............................................. 50
4. Elastic Waves in Solids ...................................... 55
4.1 Free Waves in a Homogeneous Isotropic Medium ............ 55
4.1.1 Longitudinal and Transverse Waves .................. 55
4.1.2 Boundary Conditions for Elastic Waves ............... 58
4.2 Wave Reflection at a Stress-Free Boundary. . . . . . . . . . . . . . . . . . 59
4.2.1 Boundary Conditions .............................. 59
4.2.2 Reflection of a Horizontally Polarized Wave. . . . . . . . . . . 60
4.2.3 The Reflection of Vertically Polarized Waves .......... 61
4.2.4 Particular Cases of Reflection ....................... 63
4.2.5 Inhomogeneous Waves............................. 64
4.3 Surface Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1 The Rayleigh Wave ................................ 65
4.3.2 The Surface Love Wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.3 Some Features of Love's Waves. . .. . . .. . . . . . .. .. . ... . 68
4.4 Exercises .............................................. 70
5. Waves in Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1 Classification of Waves .................................. 75
5.1.1 Dispersion Relations ............................... 75
5.1.2 Symmetric and Asymmetric Modes ................... 76
5.1.3 Cut-Off Frequencies of the Modes.. . . .. . . ..... .. . .. . . 77
5.1.4 Some Special Cases ................................ 79
5.2 Normal Modes of the Lowest Order ........................ 79
5.2.1 Quasi-Rayleigh Waves at the Plate's Boundaries. . . . . . . . 79
5.2.2 The Young and Bending Waves ...................... 81
5.3 Equations Describing the Bending of a Thin Plate ............ 82
5.3.1 Thin Plate Approximation .......................... 82
Contents IX
5.3.2 Sophie Germain Equation. ... .. . . . . .. ... . . .. . . . . .. . . 84
5.3.3 Bending Waves in a Thin Plate. . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Exercises .............................................. 86
Part II Fluid Mechanics
6. Basic Laws ofIdeal Fluid Dynamics ........................... 92
6.1 Kinematics of Fluids . ... . . ... . .... . . . . . ... ... . ... . ... .. .. 92
6.1.1 Eulerian and Lagrangian Representations of Fluid Motion 92
6.1.2 Transition from One Representation to Another. . . . . . . . 92
6.1.3 Convected and Local Time Derivatives... . ... . ... . . . .. 93
6.2 System of Equations of Hydrodynamics .................... 95
6.2.1 Equation of Continuity ............................. 95
6.2.2 The Euler Equation ................................ 95
6.2.3 Completeness of the System of Equations. . . . . . . . . . . . . . 96
6.3 The Statics of Fluids ..................................... 98
6.3.1 BasicEquations ................................... 98
6.3.2 Hydrostatic Equilibrium. Vaisala Frequency ........... 99
6.4 Bernoulli's Theorem and the Energy Conservation Law ....... 100
6.4.1 Bernoulli's Theorem ............................... 100
6.4.2 Some Applications of Bernoulli's Theorem ............ 102
6.4.3 The Bernoulli Theorem as a Consequence of the
Energy-Conservation Law .......................... 103
6.4.4 Energy Conservation Law in the General Case of
Unsteady Flow .................................... 104
6.5 Conservation of Momentum .............................. 106
6.5.1 The Specific Momentum Flux Tensor ................. 106
6.5.2 Euler's Theorem. . . . .. . . .. . . . . . . ... . .. . .. . . .. . . . . .. 107
6.5.3 Some Applications of Euler's Theorem.. . . ... . .. . . . . .. 109
6.6 Vortex Flows ofIdeal Fluids. . .. . . . . ... .. . .. . . .. . ... . . . . . . 110
6.6.1 The Circulation of Velocity .......................... 110
6.6.2 Kelvin's Circulation Theorem. .. . . . . . . . . . . . . .. . . . .. . . 111
6.6.3 Helmholtz Theorems ............................... 112
6.7 Exercises .............................................. 113
7. Potential Flow ............................................. 121
7.1 Equations for a Potential Flow ............................ 121
7.1.1 VelocityPotential .................................. 121
7 .1.2 Two-Dimensional Flow. Stream Function ............. 122
7.2 Applications of Analytical Functions to Problems
of Hydrodynamics ...................................... 124
7.2.1 The Complex Flow Potential ........................ 124
7.2.2 Some Examples of Two-Dimensional Flows. . . . . . . . . . . . 125
X Contents
7.2.3 Conformal Mapping ............................... 127
7.3 Steady Flow Around a Cylinder ........................... 129
7.3.1 Application of Conformal Mapping .................. 129
7.3.2 The Pressure Coefficient. . . . . . ... . .. . . .. . . . . .. .. . . .. 131
7.3.3 The Paradox of d' Alembert and Euler ................ 132
7.3.4 The Flow Around a Cylinder with Circulation .......... 133
7.4 Irrotational Flow Around a Sphere ........................ 134
7.4.1 The Flow Potential and the Particle Velocity ........... 134
7.4.2 The Induced Mass ................................. 136
7.5 Exercises .............................................. 137
8. Flows of Viscous Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.1 Equations of Flow of Viscous Fluid. . . .... ... . .... . ... ... .. 145
8.1.1 Newtonian Viscosity and Viscous Stresses ............. 145
8.1.2 The Navier-Stokes Equation. . . . . . . . . . . . . . . . . . . . . . . . . 146
8.1.3 The Viscous Force ................................. 148
8.2 Some Examples of Viscous Fluid Flow. . . . . . . . . . . . . . . . . . . . . . 149
8.2.1 Couette Flow ..................................... 149
8.2.2 Plane Poiseuille Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.2.3 Poiseuille Flow in a Cylindrical Pipe .................. 151
8.2.4 Viscous Fluid Flow Around a Sphere. . . . . . . . . . . . . . . . . . 153
8.2.5 Stokes' Formula for Drag ........................... 154
8.3 Boundary Layer ........................................ 156
8.3.1 Viscous Waves .................................... 157
8.3.2 The Boundary Layer. Qualitative Considerations ....... 157
8.3.3 Prandl's Equation for a Boundary Layer .............. 160
8.3.4 Approximate Theory of a Boundary Layer in a Simple
Case. .. . . . . .. . . .. . .. . . . . . . . ... ... . ... . . . . . .. . . .. . 162
8.4 Exercises .............................................. 163
9. Elements of the Theory of Turbulence ......................... 169
9.1 Qualitative Considerations. Hydrodynamic Similarity ........ 169
9.1.1 Transition from a Laminar to Turbulent Flow ... " . .. . . 169
9.1.2 Similar Flows ..................................... 171
9.1.3 Dimensional Analysis and Similarity Principle. . . . . . . . . . 173
9.1.4 Flow Around a Cylinder at Different Re . . . . . . . . . . . . . . . 174
9.2 Statistical Description of Turbulent Flows .................. 177
9.2.1 Reynolds' Equation for Mean Flow... . ... . . .. . .. . ... . 177
9.2.2 Turbulent Viscosity ................................ 180
9.2.3 Turbulent Boundary Layer .......................... 181
9.3 Locally Isotropic Turbulence ............................. 184
9.3.1 Properties of Developed Turbulence.................. 184
9.3.2 Statistical Properties of Locally Isotropic Turbulence . . . . 186
9.3.3 Kolmogorov's Similarity Hypothesis ........ '" " . . .. . 190
9.4 Exercises .............................................. 191
Contents XI
10. Surface and Internal Waves in Fluids ......................... 197
10.1 Linear Equations for Waves in Stratified Fluids. . . . . . . . . . . . 197.
10.1.1 Linearization of the Hydrodynamic Equations ...... 197
10.1.2 Linear Boundary Conditions ..................... 198
10.1.3 Equations for an Incompressible Fluid ............. 199
10.2 Surface Gravity Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10.2.1 Basic Equations ................................ 200
10.2.2 Harmonic Waves ............................... 201
10.2.3 Shallow-and Deep-Water Approximations ......... 204
10.2.4 Wave Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.3 Capillary Waves ...................................... 207
10.3.1 "Pure"CapillaryWaves ......................... 207
10.3.2 Gravity-Capillary Surface Waves. . . . . . . . . . . . . . . . . . 208
10.4 InternalGravityWaves ................................ 210
10.4.1 Introductory Remarks. . . . . . . . . . . . . . . . . . . . . . . . . .. 210
10.4.2 Basic Equation for Internal Waves.
Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . . . . 211
10.4.3 Waves in an Unlimited Medium ................... 212
10.5 Guided Propagation ofInternal Waves ................... 216
10.5.1 Qualitative Analysis of Guided Propagation ........ 216
10.5.2 Simple Model of an Oceanic Waveguide. . . . . . . . . . .. 218
10.5.3 Surface Mode. "Rigid Cover" Condition ........... 220
10.5.4 Internal Modes ................................. 221
10.6 Exercises ............................................ 224
11. Waves in Rotating Fluids ................................... 236
11.1 Inertial (Gyroscopic) Waves ............................ 236
11.1.1 The Equation for Waves in a Homogeneous
Rotating Fluid ................................. 236
11.1.2 Plane Harmonic Inertial Waves ................... 237
11.1.3 Waves in a Fluid Layer. Application to Geophysics. .. 239
11.2 Gyroscopic-Gravity Waves ............................. 241
11.2.1 General Equations. The Simplest Model of a Medium 241
11.2.2 Classification of Wave Modes .................... 242
11.2.3 Gyroscopic-Gravity Waves in the Ocean. . . . . . . . . . .. 243
11.3 TheRossbyWaves .................................... 246
11.3.1 The Tangent of f3-Plane Approximation ............ 246
11.3.2 The Barotropic Rossby Waves .................... 247
11.3.3 Joint Discussion of Stratification and the p-Effect ... 248
11.3.4 The Rossby Waves in the Ocean. . . . . . . . . . . . . . . . . .. 251
11.4 Exercises ............................................ 253
12. Sound Waves ............................................. 262
12.1 Plane Waves in Static Fluids ............................ 262
12.1.1 The System of Linear Acoustic Equations .......... 262
12.1.2 Plane Waves ................................... 263