Computational Atmospheric Acoustics

330 Pages·2001·12.55 MB·English

by  Erik M. Salomons (auth.)

computational atmospheric acoustics
computational atmospheric acoustics
by
ERIK M. SALOMONS
TNO Institute ofA pplied Physics,
Delft, The Netherlands
SPRINGER -SCIENCE+BUSINESS MEDIA,B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-0390-5 ISBN 978-94-010-0660-6 (eBook)
DOl 10.1007/978-94-010-0660-6
Printed on acidjree paper
All Rights Reserved
© 200 I Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 200 I
Softcover reprint of the hardcover 1st edition 2001
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
preface xi
1 Introduction 1
1.1 Atmospheric acoustics 1
1.2 Scope of the book. . . 2
1.3 Structure of the book 3
2 Unbounded homogeneous atmosphere 5
2.1 Introduction............... 5
2.2 Plane waves . . . . . . . . . . . . . . . 6
2.3 Complex notation for harmonic waves 8
2.4 Spherical waves . . . . . . . . . . . 9
2.5 Atmospheric absorption . . . . . . 9
2.6 Sound pressure level and spectrum 11
3 Homogeneous atmosphere above a ground surface 21
3.1 Introduction....................... 21
3.2 Reflection of spherical waves by a ground surface . . 21
3.3 Spherical-wave reflection coefficient and ground impedance 23
3.4 Relative sound pressure level 25
3.5 Examples . . . . . . . 27
4 Atmospheric refraction 37
4.1 Introduction...... 37
4.2 Atmospheric refraction . 39
4.3 Effective sound speed. 40
4.4 Ray model. . . . . . . 42
4.5 FFP and PE methods 48
4.6 Examples . . . . . . . 52
4.6.1 Moving-medium effects 53
4.6.2 Angular limitation of the PE method 56
4.6.3 Accuracy of the axisymmetric approximation 56
4.6.4 Atmospheric refraction. . . 57
4.6.5 Accuracy of the ray model. 58
v
vi
5 Atmospheric turbulence 67
5.1 Introduction.............. 67
5.2 Non-refracting turbulent atmosphere 67
5.3 Refracting turbulent atmosphere 68
5.4 Examples . . . . . . . . . . . . . 70
6 Irregular terrain 77
6.1 Introduction........... 77
6.2 Hills and other terrain profiles. 77
6.3 Examples . . . . . . . . . . . . 79
7 Noise barriers 85
7.1 Introduction ........ . 85
7.2 Non-refracting atmosphere. 85
7.3 Refracting atmosphere 86
704 Examples ......... . 87
A Basic acoustic equations for a homogeneous atmosphere 91
A.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . 91
A.2 The linear acoustic equations and the wave equation . . . . 91
A.3 Helmholtz equation for harmonic waves . . . . . . . . . . . 94
AA Inhomogeneous Helmholtz equation for a harmonic monopole source 95
B Free field of a point source 99
B.1 Introduction .................... . 99
B.2 Acoustic power of a source ............ . 99
B.3 Sound pressure level and geometrical attenuation 100
BA Spectral decomposition. 102
B.5 Atmospheric absorption 108
B.6 Doppler effect ... 111
C Acoustic impedance 113
C.1 Introduction .......... . 113
C.2 Impedance of a ground surface 113
C.3 Impedance of air ....... . 116
C.4 Impedance of porous media . . 117
C.5 Normal reflection by a ground surface 120
C.6 Normal reflection by a layered ground 121
D Reflection of sound waves 123
D.l Introduction ...... . 123
D.2 Reflection of plane waves " 123
D.3 Local reaction approximation 126
DA Reflection of spherical waves. 129
D 04.1 Locally reacting ground surface . 129
DA.2 Extended reacting ground surface. 135
vii
E Basic acoustic equations for a layered refracting atmosphere 139
E.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
E.2 Moving atmosphere .......................... 140
E.2.1 Helmholtz equation in the horizontal wave number domain 140
E.2.2 Alternate derivation . . . . . . . . . . . . . . . 144
E.2.3 Helmholtz equation in the spatial domain . . . 144
E.3 Non-moving atmosphere with an effective sound speed 145
E.4 Axisymmetric approximation . . . . . . . 146
E.5 Alternate approach . . . . . . . . . . . . . 150
E.6 Representation of atmospheric absorption 150
F Generalized Fast Field Program (FFP) 153
F.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 153
F.2 Solution of the Helmholtz equation . . . . . . . . . . . . . 154
F.3 Extrapolation from the ground and the top to the source 156
F.4 Field at the receiver . . . . . . . . . 157
F.5 Deformation of the integration path 158
F.6 Improvement of numerical accuracy. 159
F.7 Analytical example. . . . . . . . 160
G Parabolic Equation (PE) method 163
G.1 Introduction. . . . . . . . . . . . 163
G.2 Basic approach of the CNPE method. 164
G.3 Narrow-angle parabolic equation . . . 165
G.4 Alternate derivation of the narrow-angle parabolic equation 166
G.5 Wide-angle parabolic equation ................ 167
G.6 Finite-difference solution of the narrow-angle parabolic equation 168
G.7 Finite-difference solution of the wide-angle parabolic equation. 170
G.8 Boundary condition at the ground surface 171
G.9 Boundary condition at the top of the grid 172
G.1O Density profile ..... 172
G.11 Finite-element solution . . . . . . . . . . 174
G.12Startingfield .. . . . . . . . . . . . . . 175
G.12.1 Narrow-angle parabolic equation 176
G.12.2 Wide-angle parabolic equation . 177
G.I2.3 Source near a finite-impedance ground surface 179
H Green's Function Parabolic Equation (GFPE) method 181
H.I Introduction. . . . . . . . . . . . . . . 181
H.2 Unbounded non-refracting atmosphere . . . . . . . . . . 181
H.3 Kirchhoff-Helmholtz integral equation . . . . . . . . . . 183
H.4 General Green's function approach to wave propagation 186
H.5 Non-refracting atmosphere. 187
H.6 Refracting atmosphere 191
H.7 Alternate derivation . . . . 193
viii
H.8 Relation to the Fourier split-step method 193
H.9 Alternate refraction factor. . . . . . . 194
H.lO Starting field . . . . . . . . . . . . . . 195
H.ll Discretization of the Fourier integrals. 197
H.12 Three-dimensional GFPE method. 199
I Atmospheric turbulence 203
1.1 Introduction................. 203
1.2 Turbulence in sound propagation models . 204
1.3 Reynolds number and onset of turbulence 205
104 Random fields. . . . 207
1.5 The 'two-thirds law' . . . . . . . . . . . . 209
1.6 Spectral density . . . . . . . . . . . . . . . 210
1.7 Gaussian, Kolmogorov, and von Karman spectra 211
1.7.1 Atmosphere with temperature fluctuations. 212
1.7.2 Atmosphere with wind and temperature fluctuations 215
1.8 Limitations of the statistical description of turbulence 217
J Atmospheric turbulence in the PE method 221
J.l Introduction............................. 221
J.2 Turbulent phase factor in the PE method . . . . . . . . . . .. 222
J.3 Random realizations of the field of refractive-index fluctuations 224
J.3.1 Refractive-index fluctuations in the CNPE method 226
J.3.2 Refractive-index fluctuations in the GFPE method 226
J.3.3 Numerical parameters . . . . . . . . . . . . . 226
Jo4 Turbulence in the three-dimensional GFPE method. . . . 228
K Analytical model for a non-refracting turbulent atmosphere 231
K.l Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 231
K.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
K.3 Coherence factor for Gaussian and von Karman spectra 235
Ko4 Axisymmetric turbulence ................. 236
L Ray model including caustic diffraction fields 239
L.l Introduction.... 239
L.2 Setup of the model . . . . . 240
L.2.1 System ....... 240
L.2.2 Sound pressure field 240
L.3 Geometrical acoustics solution. 242
L.3.1 Ray paths. . . . . . . 243
L.3.2 Caustic curves .... 246
L.3.3 Indices of caustic rays 249
L.304 Ground reflections 251
L.3.5 Phases of the rays 252
L.3.6 Focusing factors . 253
ix
L.4 Caustic diffraction fields . . . . . . . . . . . . 254
L.4.l Theory of caustics . . . . . . . . . . . 254
L.4.2 Extrapolation into the shadow region 257
L.4.3 Application of caustic theory in the ray model 258
L.5 Effects of atmospheric turbulence . . . . . . . . 260
M Computational methods for irregular terrain 263
M.l Introduction. . . . . . . . . . . . . . . . . 263
M.2 The conformal mapping method ..... 264
M.3 Generalized Terrain PE (GTPE) method. 267
M.3.l Terrain following coordinates . . . 267
M.3.2 Transformation of the Helmholtz equation . 268
M.3.3 First-order GTPE .... 269
M.3.4 Second-order GTPE . . . . . . . . . . . . . 269
M.3.5 Finite-difference solution. . . . . . . . . . . 271
M.3.6 Boundary conditions at the ground and the top . 272
N Wind and temperature profiles in the atmosphere 279
N.l Introduction. . . . . . . . . . . . . . . . . . . . . . . 279
N.2 Boundary layer and surface layer of the atmosphere 280
N.3 Potential temperature . . . . . 281
N.4 Mean and turbulent parts . . . 282
N.5 Heat flux and momentum flux . 282
N.6 Similarity relations . . . . . . . 283
o
Sound propagation over a screen 289
0.1 Introduction. . . . . . . . . . . . . . . . . . . . . . 289
0.2 Analytical model for a non-refracting atmosphere. 290
0.3 PE method for a refracting atmosphere 292
0.4 Wind field near a screen . . . . . . . . . . . . . . . 293
P The method of stationary phase 297
References 301
List of symbols 314
Index 327
preface
This book is intended for anyone who is interested in the computation of sound
propagation in the atmosphere. In some simple cases the computation can
be performed analytically. In most cases, however, the computation must be
performed numerically, as the atmosphere is a complex medium for sound waves.
The book describes current computational methods for sound propagation in the
atmosphere.
The book is based on many excellent articles from the literature of atmo
spheric acoustics. Articles presented at the International Symposia on Long
Range Sound Propagation have been particularly valuable.
The book was written 'on week-ends', but the inspiring atmosphere on
week-days at the TNO Institute of Applied Physics has very much contributed
to the book.
The author is grateful to Andrew Thean and Niels Salomons for many valu
able comments on the text. Above all, the author is grateful to Marga Salomons,
Michelle Salomons, and Lisa Salomons for help and support in writing the book.
Erik M. Salomons
TNO Institute of Applied Physics
May 2001
xi
Chapter 1
Introduction
1.1 Atmospheric acoustics
Atmospheric acoustics is the science of sound propagation in the atmosphere.
The basic geometry with a source and a receiver is illustrated in Fig. 1.1. Sound
waves are generated by a source and travel through the atmosphere to a receiver.
The source may be a whistling bird, as in Fig. 1.1. Other important examples
of sources are cars, trains, and airplanes.
The atmosphere is a complex medium for sound propagation. Wind and
temperature distributions in the atmosphere play an important role in the prop
agation. The influence of wind is illustrated by the fact that the sound from a
source near the ground is often louder on the downwind side of the source than
on the upwind side of the source. Sound propagation is affected not only by the
distributions of the mean wind and temperature, but also by rapid fluctuations
of wind and temperature, i.e. atmospheric turbulence.
The ground surface can be considered as the boundary of the propagation
medium. Reflection of sound waves by the ground surface plays an impor
tant role in the propagation. One distinguishes hard grounds and absorbing
receiver
source
atmosphere
ground
Figure 1.1. Basic geometry of sound propagation in the atmosphere.
1
E. M. Salomons, Computational Atmospheric Acoustics
© Springer Science+Business Media Dordrecht 2001