Theory of nonlinear acoustics in fluids

The aim of the present book is to present theoretical nonlinear aco- tics with equal stress on physical and mathematical foundations. We have attempted explicit and detailed accounting for the physical p- nomena treated in the book, as well as their modelling, and the f- mulation and solution of the mathematical models. The nonlinear acoustic phenomena described in the book are chosen to give phy- cally interesting illustrations of the mathematical theory. As active researchers in the mathematical theory of nonlinear acoustics we have found that there is a need for a coherent account of this theory from a unified point of view, covering both the phenomena studied and mathematical techniques developed in the last few decades. The most ambitious existing book on the subject of theoretical nonlinear acoustics is "Theoretical Foundations of Nonlinear Aco- tics" by O. V. Rudenko and S. I. Soluyan (Plenum, New York, 1977). This book contains a variety of applications mainly described by Bu- ers' equation or its generalizations. Still adhering to the subject - scribed in the title of the book of Rudenko and Soluyan, we attempt to include applications and techniques developed after the appearance of, or not included in, this book. Examples of such applications are resonators, shockwaves from supersonic projectiles and travelling of multifrequency waves. Examples of such techniques are derivation of exact solutions of Burgers' equation, travelling wave solutions of Bu- ers' equation in non-planar geometries and analytical techniques for the nonlinear acoustic beam (KZK) equation.

Preface	p. xi
1 Introduction	p. 1
1.1 The place of acoustics in fluid mechanics	p. 1
1.2 Nonlinear acoustics before 1950	p. 2
1.3 Special phenomena in nonlinear acoustics	p. 4
1.3.1 Common theoretical description of nonlinear acoustics phenomena	p. 4
1.3.2 Generation and propagation of higher harmonics in travelling waves	p. 5
1.3.3 Generation and propagation of combination frequency travelling waves	p. 7
1.3.4 Propagation of travelling short pulses and N-waves	p. 8
1.3.5 Propagation of limited sound beams	p. 9
1.3.6 Waves in closed tubes	p. 9
2 Physical theory of nonlinear acoustics	p. 11
2.1 Basic theory of motion of a diffusive medium	p. 12
2.1.1 Conservation of mass; the continuity equation	p. 13
2.1.2 Conservation of momentum. Navier-Stokes equations	p. 14
2.1.3 Conservation of energy	p. 15
2.1.4 Ideal fluid equation of state	p. 18
2.2 Derivation of the three dimensional wave equation of nonlinear acoustics (Kuznetsov's equation)	p. 20
2.3 Wave equations of nonlinear acoustics	p. 24
2.3.1 Burgers' equation	p. 24
2.3.2 Generalized Burgers' equation	p. 27
2.3.3 The KZK equation	p. 28
3 Basic methods of nonlinear acoustics	p. 31
3.1 Solution methods to the Riemann wave equation	p. 31
3.1.1 Physical interpretation of the Riemann equation	p. 31
3.1.2 Continuous wave solution	p. 33
3.1.3 Shock wave solution	p. 35
3.1.4 Rule of equal areas	p. 38
3.1.5 Prediction of wave behaviour from area differences	p. 42
3.2 Exact solution of Burgers' equation	p. 45
3.2.1 The Cole-Hopf solution of Burgers' equation	p. 46
3.2.2 Burgers' equation with vanishing diffusivity	p. 49
4 Nonlinear waves with zero and vanishing diffusion	p. 53
4.1 Short pulses	p. 53
4.1.1 Triangular pulses	p. 53
4.1.2 N-waves	p. 56
4.2 Sinusoidal waves	p. 59
4.2.1 Continuous solution	p. 59
4.2.2 The Bessel-Fubini solution	p. 61
4.2.3 Sawtooth solution	p. 62
4.2.4 The one saddle-point method	p. 66
4.2.5 Time reversal	p. 72
4.3 Modulated Riemann waves	p. 76
4.3.1 Direct method for bifrequency boundary condition	p. 76
4.3.2 The one saddle-point method for bifrequency boundary condition	p. 79
4.3.3 Characteristic multifrequency waves	p. 84
5 Nonlinear plane diffusive waves	p. 93
5.1 Planar N-waves	p. 93
5.1.1 Shock solution	p. 93
5.1.2 Old-age solution	p. 98
5.1.3 The old-age solution found by an alternative method	p. 100
5.2 Planar harmonic waves. The Fay solution	p. 105
5.2.1 Derivation of Fay's solution from the Cole-Hopf solution	p. 105
5.2.2 Direct derivation of Fay's solution	p. 109
5.2.3 Proof that Fay's solution satisfies Burgers' equation	p. 110
5.2.4 Some notes on Fay's solution	p. 112
5.3 Planar harmonic waves. The Khokhlov-Soluyan solution	p. 114
5.3.1 Derivation of the Khokhlov-Soluyan solution	p. 114
5.3.2 Comparison between the Fay and the Khokhlov-Soluyan solutions	p. 118
5.3.3 Comparison between the Khokhlov-Soluyan solution and the sawtooth solution	p. 122
5.4 Planar harmonic waves. The exact solution	p. 125
5.4.1 Recursion formulae for the Fourier series of the exact solution	p. 125
5.4.2 Solving recursion formulae by discrete integration	p. 129
5.4.3 Comparison of Fourier coefficients in the Bessel-Fubini solution, the Fay solution and the exact solution	p. 133
5.5 Multifrequency waves	p. 137
5.5.1 Expressions for multifrequency solutions	p. 137
5.5.2 Bifrequency solutions and creation of combination frequencies	p. 141
6 Nonlinear cylindrical and spherical diffusive waves	p. 149
6.1 Dimensionless generalized Burgers' equations	p. 150
6.2 Cylindrical N-waves	p. 153
6.2.1 Evolution of an initial cylindrical N-wave	p. 153
6.2.2 Four-step procedure for finding the asymptotic solution	p. 154
6.3 The decay of a shockwave from a supersonic projectile	p. 166
6.3.1 Linear theory of the wave from a supersonic projectile	p. 167
6.3.2 Nonlinear theory of the wave from a supersonic projectile	p. 173
6.4 Periodic cylindrical and spherical waves	p. 186
6.4.1 Spherical periodic waves	p. 187
6.4.2 Cylindrical periodic waves	p. 193
7 Nonlinear bounded sound beams	p. 199
7.1 The KZK equation	p. 201
7.1.1 Dimensionless KZK equation	p. 201
7.1.2 Transformation of the KZK equation to a generalized Burgers' equation	p. 205
7.1.3 Expansion of the solution around the center of the beam	p. 205
7.1.4 Solution for a circular beam	p. 208
7.2 Propagation of a shock wave in a sound beam	p. 210
7.2.1 Determination of the boundary condition from the series solution	p. 210
7.2.2 Solution of generalized Burgers' equation	p. 214
7.2.3 Conditions for shock preservation	p. 216
8 Nonlinear standing waves in closed tubes	p. 219
8.1 Nonlinear and dissipative effects at non-resonant and resonant driving frequencies	p. 221
8.1.1 Linear theory of standing waves	p. 222
8.1.2 Discussion of the small numbers in the problem of nonlinear standing waves	p. 224
8.2 Equations of nonlinear standing waves	p. 226
8.2.1 Perturbation solution and boundary conditions of Kuznetsov's equation	p. 226
8.2.2 Equations of resonant standing waves	p. 230
8.3 Steady-state resonant vibrations in a non-dissipative medium	p. 231
8.3.1 Continuous solution	p. 232
8.3.2 Shock solution	p. 234
8.3.3 The Q-factor	p. 237
8.4 Steady-state resonant vibrations in a dissipative medium	p. 238
8.4.1 Mathieu equation solution	p. 238
8.4.2 Perturbation theory. Matching outer and inner solutions	p. 240
8.4.3 Perturbation theory. Uniform solution	p. 244
8.5 An example of velocity field in a resonator	p. 247
Bibliography	p. 251
Name index	p. 271
Subject index	p. 279

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