Theory of Nonlinear Acoustics in Fluids

289 Pages·2004·11.75 MB·English

by  Bengt O. Enflo

THEORY OF NONLINEAR ACOUSTICS IN FLUIDS
FLUID MECHANICS AND ITS APPLICATIONS
Volume 67
Series Editor: R. MOREAU
MADYLAM
Ecole Nationale Supérieure d'Hydraulique de Grenoble
Boîte Postale 95
38402 Saint Martin d'Hères Cedex, France
Aims and Scope of the Series
The purpose of this series is to focus on subjects in which fluid mechanics plays a funda-
mental role.
As well as the more traditional applications of aeronautics, hydraulics, heat and mass
transfer etc., books will be published dealing with topics which are currently in a state of
rapid development, such as turbulence, suspensions and multiphase fluids, super and hy-
personic flows and numerical modelling techniques.
It is a widely held view that it is the interdisciplinary subjects that will receive intense
scientific attention, bringing them to the forefront of technological advancement. Fluids
have the ability to transport matter and its properties as well as transmit force, therefore
fluid mechanics is a subject that is particulary open to crossfertilisation with other scien-
ces and disciplines of engineering. The subject of fluid mechanics will be highly relevant
in domains such as chemical, metallurgical, biological and ecological engineering. This
series is particularly open to such new multidisciplinary domains.
The median level of presentation is the first year graduate student. Some texts are mono-
graphs defining the current state of a field; others are accessible to final year undergradu-
ates; but essentially the emphasis is on readability and clarity.
Theory of Nonlinear Acoustics
in Fluids
by
BENGT O. ENFLO
Kungl. Tekniska Högskolan,
Department of Mechanics, Stockholm, Sweden
and
CLAES M. HEDBERG
Blekinge Tekniska Högskola,
Department of Mechanical Engineering, Karlskrona, Sweden
KLUWER ACADEMIC PUBLISHERS
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eBookISBN: 0-306-48419-6
Print ISBN: 1-4020-0572-5
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Contents
Preface xi
1 Introduction 1
1.1 The place of acoustics in fluid mechanics 1
1.2 Nonlinear acoustics before 1950 2
1.3 Special phenomena in nonlinear acoustics 4
1.3.1 Common theoretical description of nonlinear acous-
tics phenomena 4
1.3.2 Generation and propagation of higher harmon-
ics in travelling waves 5
1.3.3 Generation and propagation of combination fre-
quency travelling waves 7
1.3.4 Propagation of travelling short pulses and N-waves 8
1.3.5 Propagation of limited sound beams 9
1.3.6 Waves in closed tubes 9
2 Physical theory of nonlinear acoustics 11
2.1 Basic theory of motion of a diffusive medium 12
2.1.1 Conservation of mass; the continuity equation 13
2.1.2 Conservation of momentum. Navier-Stokes equa-
tions 14
2.1.3 Conservation of energy 15
2.1.4 Ideal fluid equation of state 18
2.2 Derivation of the three dimensional wave equation of
nonlinear acoustics (Kuznetsov’s equation) 20
2.3 Wave equations of nonlinear acoustics 24
v
vi
2.3.1 Burgers’ equation 24
2.3.2 Generalized Burgers’ equation 27
2.3.3 The KZK equation 28
3 Basic methods of nonlinear acoustics 31
3.1 Solution methods to the Riemann
wave equation 31
3.1.1 Physical interpretation of the Riemann equation 31
3.1.2 Continuous wave solution 33
3.1.3 Shock wave solution 35
3.1.4 Rule of equal areas 38
3.1.5 Prediction of wave behaviour from area differences 42
3.2 Exact solution of Burgers’ equation 45
3.2.1 The Cole-Hopf solution of Burgers’ equation 46
3.2.2 Burgers’ equation with vanishing diffusivity 49
4 Nonlinear waves with zero and vanishing diffusion 53
4.1 Short pulses 53
4.1.1 Triangular pulses 53
4.1.2 N-waves 56
4.2 Sinusoidal waves 59
4.2.1 Continuous solution 59
4.2.2 The Bessel-Fubini solution 61
4.2.3 Sawtooth solution 62
4.2.4 The one saddle-point method 66
4.2.5 Time reversal 72
4.3 Modulated Riemann waves 76
4.3.1 Direct method for bifrequency boundary condition 76
4.3.2 The one saddle-point method for bifrequency
boundary condition 79
4.3.3 Characteristic multifrequency waves 84
5 Nonlinear plane diffusive waves 93
5.1 Planar N-waves 93
5.1.1 Shock solution 93
5.1.2 Old-age solution 98
vii
5.1.3 The old-age solution found by an alternative
method 100
5.2 Planar harmonic waves. The Fay solution 105
5.2.1 Derivation of Fay’s solution from the Cole-Hopf
solution 105
5.2.2 Direct derivation of Fay’s solution 109
5.2.3 Proof that Fay’s solution satisfies Burgers’ equa-
tion 110
5.2.4 Some notes on Fay’s solution 112
5.3 Planar harmonic waves. The Khokhlov-Soluyan solution 114
5.3.1 Derivation of the Khokhlov-Soluyan solution 114
5.3.2 Comparison between the Fay and the Khokhlov-
Soluyan solutions 118
5.3.3 Comparison between the Khokhlov-Soluyan so-
lution and the sawtooth solution 122
5.4 Planar harmonic waves. The exact solution 125
5.4.1 Recursion formulae for the Fourier series of the
exact solution 125
5.4.2 Solving recursion formulae by discrete integration129
5.4.3 Comparison of Fourier coefficients in the Bessel-
Fubini solution, the Fay solution
and the exact solution 133
5.5 Multifrequency waves 137
5.5.1 Expressions for multifrequency solutions 137
5.5.2 Bifrequency solutions and creation of combina-
tion frequencies 141
6 Nonlinear cylindrical and spherical diffusive waves 149
6.1 Dimensionless generalized Burgers’ equations 150
6.2 Cylindrical N-waves 153
6.2.1 Evolution of an initial cylindrical N-wave 153
6.2.2 Four-step procedure for finding the asymptotic
solution 154
6.3 The decay of a shockwave from a supersonic projectile 166
6.3.1 Linear theory of the wave from a supersonic pro-
jectile 167
viii
6.3.2 Nonlinear theory of the wave from a supersonic
projectile 173
6.4 Periodic cylindrical and spherical waves 186
6.4.1 Spherical periodic waves 187
6.4.2 Cylindrical periodic waves 193
7 Nonlinear bounded sound beams 199
7.1 The KZK equation 201
7.1.1 Dimensionless KZK equation 201
7.1.2 Transformation of the KZK equation to a gen-
eralized Burgers’ equation 205
7.1.3 Expansion of the solution around the center of
the beam 205
7.1.4 Solution for a circular beam 208
7.2 Propagation of a shock wave in a sound beam 210
7.2.1 Determination of the boundary condition from
the series solution 210
7.2.2 Solution of generalized Burgers’ equation 214
7.2.3 Conditions for shock preservation 216
8 Nonlinear standing waves in closed tubes 219
8.1 Nonlinear and dissipative effects at non-resonant and
resonant driving frequencies 221
8.1.1 Linear theory of standing waves 222
8.1.2 Discussion of the small numbers in the problem
of nonlinear standing waves 224
8.2 Equations of nonlinear standing waves 226
8.2.1 Perturbation solution and boundary conditions
of Kuznetsov’s equation 226
8.2.2 Equations of resonant standing waves 230
8.3 Steady-state resonant vibrations in a non-dissipative
medium 231
8.3.1 Continuous solution 232
8.3.2 Shock solution 234
8.3.3 The Q-factor 237
8.4 Steady-state resonant vibrations in a dissipative medium238
ix
8.4.1 Mathieu equation solution 238
8.4.2 Perturbation theory. Matching outer and inner
solutions 240
8.4.3 Perturbation theory. Uniform solution 244
8.5 An example of velocity field in a resonator 247
Bibliography 251
Name index 271
Subject index 279
Preface
The aim of the present book is to present theoretical nonlinear acous-
tics with equal stress on physical and mathematical foundations. We
have attempted explicit and detailed accounting for the physical phe-
nomena treated in the book, as well as their modelling, and the for-
mulation and solution of the mathematical models. The nonlinear
acoustic phenomena described in the book are chosen to give physi-
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 Acous-
tics” by O.V. Rudenko and S.I. Soluyan (Plenum, New York, 1977).
This book contains a variety of applications mainly described by Burg-
ers’ equation or its generalizations. Still adhering to the subject de-
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 Burg-
ers’ equation in non-planar geometries and analytical techniques for
the nonlinear acoustic beam (KZK) equation. The analytical tech-
niques are developed from first principles, i.e. from the continuity
equation, Navier-Stokes’ equations, the heat conduction equation and
the constitutive equation of the medium, in which the nonlinear acous-
xi