Theory of Nonlinear Acoustics in... | Buy

This book presents theoretical nonlinear acoustics in fluids with equal stress on physical foundations and mathematical methods. From first principles in fluid mechanics and thermodynamics a universal mathematical model (Kuznetsov's equation) of nonlinear acoustics is developed. This model is applied to problems such as nonlinear generation of higher harmonics and combination frequencies, the shockwave from a supersonic projectile, propagation of shocks in acoustic beams and nonlinear standing waves in resonators.Special for the book is the coherent account of nonlinear acoustic theory from a unified point of view and the detailed presentations of the mathematical techniques for solving the nonlinear acoustic model equations. The book differs from mathematical books on nonlinear wave equations by its stress on their origin in physical principles and their use for physical applications. It differs from books on applications of nonlinear acoustics by its ambition to explain all steps in mathematical derivations of physical results. It is useful for practicians and researchers in acoustics feeling the need for more theoretical understanding. It can be used as a textbook for graduate or advanced undergraduate students with an adequate background in physics and mathematical analysis, specializing in acoustics, mechanics or applied mathematics.

Preface

xi

Introduction

1

(10)

The place of acoustics in fluid mechanics

1

(1)

Nonlinear acoustics before 1950

2

(2)

Special phenomena in nonlinear acoustics

4

(7)

Common theoretical description of nonlinear acoustics phenomena

4

(1)

Generation and propagation of higher harmonics in travelling waves

5

(2)

Generation and propagation of combination frequency travelling waves

7

(1)

Propagation of travelling short pulses and N-waves

8

(1)

Propagation of limited sound beams

9

(1)

Waves in closed tubes

9

(2)

Physical theory of nonlinear acoustics

11

(20)

Basic theory of motion of a diffusive medium

12

(8)

Conservation of mass; the continuity equation

13

(1)

Conservation of momentum. Navier-Stokes equations

14

(1)

Conservation of energy

15

(3)

Ideal fluid equation of state

18

(2)

Derivation of the three dimensional wave equation of nonlinear acoustics (Kuznetsov's equation)

20

(4)

Wave equations of nonlinear acoustics

24

(7)

Burgers' equation

24

(3)

Generalized Burgers' equation

27

(1)

The KZK equation

28

(3)

Basic methods of nonlinear acoustics

31

(22)

Solution methods to the Riemann wave equation

31

(14)

Physical interpretation of the Riemann equation

31

(2)

Continuous wave solution

33

(2)

Shock wave solution

35

(3)

Rule of equal areas

38

(4)

Prediction of wave behaviour from area differences

42

(3)

Exact solution of Burgers' equation

45

(8)

The Cole-Hopf solution of Burgers' equation

46

(3)

Burgers' equation with vanishing diffusivity

49

(4)

Nonlinear waves with zero and vanishing diffusion

53

(40)

Short pulses

53

(6)

Triangular pulses

53

(3)

N-waves

56

(3)

Sinusoidal waves

59

(17)

Continuous solution

59

(2)

The Bessel-Fubini solution

61

(1)

Sawtooth solution

62

(4)

The one saddle-point method

66

(6)

Time reversal

72

(4)

Modulated Riemann waves

76

(17)

Direct method for bifrequency boundary condition

76

(3)

The one saddle-point method for bifrequency boundary condition

79

(5)

Characteristic multifrequency waves

84

(9)

Nonlinear plane diffusive waves

93

(56)

Planar N-waves

93

(12)

Shock solution

93

(5)

Old-age solution

98

(2)

The old-age solution found by an alternative method

100

(5)

Planar harmonic waves. The Fay solution

105

(9)

Derivation of Fay's solution from the Cole-Hopf solution

105

(4)

Direct derivation of Fay's solution

109

(1)

Proof that Fay's solution satisfies Burgers' equation

110

(2)

Some notes on Fay's solution

112

(2)

Planar harmonic waves. The Khokhlov-Soluyan solution

114

(11)

Derivation of the Khokhlov-Soluyan solution

114

(4)

Comparison between the Fay and the Khokhlov-Soluyan solutions

118

(4)

Comparison between the Khokhlov-Soluyan solution and the sawtooth solution

122

(3)

Planar harmonic waves. The exact solution

125

(12)

Recursion formulae for the Fourier series of the exact solution

125

(4)

Solving recursion formulae by discrete integration

129

(4)

Comparison of Fourier coefficients in the Bessel-Fubini solution, the Fay solution and the exact solution

133

(4)

Multifrequency waves

137

(12)

Expressions for multifrequency solutions

137

(4)

Bifrequency solutions and creation of combination frequencies

141

(8)

Nonlinear cylindrical and spherical diffusive waves

149

(50)

Dimensionless generalized Burgers' equations

150

(3)

Cylindrical N-waves

153

(13)

Evolution of an initial, cylindrical N-wave

153

(1)

Four-step procedure for finding the asymptotic solution

154

(12)

The decay of a shockwave from a supersonic projectile

166

(20)

Linear theory of the wave from a supersonic projectile

167

(6)

Nonlinear theory of the wave from a supersonic projectile

173

(13)

Periodic cylindrical and spherical waves

186

(13)

Spherical periodic waves

187

(6)

Cylindrical periodic waves

193

(6)

Nonlinear bounded sound beams

199

(20)

The KZK equation

201

(9)

Dimensionless KZK equation

201

(4)

Transformation of the KZK equation to a generalized Burgers' equation

205

(1)

Expansion of the solution around the center of the beam

205

(3)

Solution for a circular beam

208

(2)

Propagation of a shock wave in a sound beam

210

(9)

Determination of the boundary condition from the series solution

210

(4)

Solution of generalized Burgers' equation

214

(2)

Conditions for shock preservation

216

(3)

Nonlinear standing waves in closed tubes

219

(32)

Nonlinear and dissipative effects at non-resonant and resonant driving frequencies

221

(5)

Linear theory of standing waves

222

(2)

Discussion of the small numbers in the problem of nonlinear standing waves

224

(2)

Equations of nonlinear standing waves

226

(5)

Perturbation solution and boundary conditions of Kuznetsov's equation

226

(4)

Equations of resonant standing waves

230

(1)

Steady-state resonant vibrations in a non-dissipative medium

231

(7)

Continuous solution

232

(2)

Shock solution

234

(3)

The Q-factor

237

(1)

Steady -state resonant vibrations in a dissipative medium

238

(9)

Mathieu equation solution

238

(2)

Perturbation theory Matching outer and inner solutions

240

(4)

Perturbation theory. Uniform solution

244

(3)

An example of velocity field in a resonator

247

(4)

Bibliography

251

(20)

Name index

271

(8)

Subject index

279

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