Table Of ContentFormulas of Acoustics
F. P. Mechel (Ed.)
Formulas of Acoustics
With contributions by
M.1. Munjal . M. VorHinder . P. Koitzsch . M. Ochmann
A. Cummings· W. Maysenholder . W. Arnold
o.
V. Rudenko
1s t ed. 2002. Corr. 2nd printing
With approximate 620 Figures and 70 Tables
~ Springer
Editor:
Prof. Dr. Fridolin P. Mechel
Landhausstr. 12
71120 Grafenau
Germany
ISBN 978-3-662-07298-1 ISBN 978-3-662-07296-7 (eBook)
DOI 10.1007/978-3-662-07296-7
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v
Formulas of Acoustics Editorial
A
Editorial
A.1 Preface :
Modem acoustics is more and more based on computations, and computations are based on
formulas. Such work needs previous and contemporary results. It consumes much time and
effort to search needed formulas during the actual work. Therefore, fundamentals and results
of acoustics that can be expressed as formulas will be collected in this book.
The author has compiled a private formula collection during all his professional life. This
personal collection gave the stimulus to expand it and to make it available to colleagues.
The present formula collection is subdivided into fields of acoustics. For some fields, in which
this author is not expert enough, he invited co-authors to contribute. Most colleagues contac
ted for possible contributions were convinced of the project and agreed spontaneously.
The material within a field of acoustics is subdivided in sections which deal with a defmed
task. Some overlap of sections has to be tolerated; but the subdivision into well-defined sec
tions will be helpful to the reader to find a particular topic of interest.
The present formula collection should not be considered a textbook in a condensed form. Deri
vations of a presented result will be described only as far as they are helpful in understanding
the problem; the more interested reader is referred to the "source" of the result. Useful prin
ciples and computational procedures will also be included, even if they need more description.
Symbols and quantities will be well defined, and wherever useful a sketch will help to explain
the object and the task.
One of the advantages of a formula collection is seen in uniform definitions, notations and
symbols for quantities. A strict uniformity in the form of a central list of symbols used never
works, according to this author's observation. Therefore, only commonly used symbols (such
as medium density, speed of sound, circular frequency, etc.) are collected in a central list of
symbols; other symbols are defined in the relevant chapter.
Many sections contain, below their title or in the text, a reference to the literature. It cannot be
the task and intention of this book either to indicate time priorities of publications concerning
Formulas of Acoustics VI Editorial
a topic or to give a survey of the existing literature. The reference quoted is the source of more
information, which the author has used.
Computation and formulas are a form of mathematics. Higher transcendental functions are
used in the mathematics of acoustics. Functions will be explained by reference to mathematical
literature, if necessary. If functions are used with different definitions in the literature, the
definition used will be presented. For more details of higher transcendental functions the
reader is referred to mathematical formula collections and handbooks.
To the author's knowledge, this is the first formula collection existing in acoustics, and, like
every first realisation of an idea, it is an experiment. Also for the publisher, who has limited
the first edition to about 1200 pages. This frame is one of the reasons why not all important
results from the literature are contained in the book; another reason is the fact that the
author(s) had no knowledge of such results. If this book will be accepted by the "acoustics
community" further editions surely will follow without page number limitation. Colleagues
who wish to find formulas in a future edition which not yet are included are kindly invited to
give suitable indications to the author.
Nevertheless the authors think that the book in its present form contains most of traditional
and modem results of both fundamental and special character so that the book can be helpful
to researchers and engineers in the fields of physical acoustics, noise control, and room
acoustics.
The manuscript was written in a camera ready form (in order to avoid proof reading). So prin
ting errors are the responsibility of the author. He would be grateful for indications of such
errors.
The author gratefully acknowledges the support given to the project by the co-authors and by
the publisher.
Grafenau, October 2001
Formulas of Acoustics VII Contents
A.2
Contents
A. Editorials
A.I Preface ............................................................................................................................... V
A.2 Contents ......................................................................................................................... VII
A.3 Conventions ................................................................................................................. XXI
B. General Linear Fluid Acoustics
B.I Fundamental differential equations ................................................................................... 1
B.2 Material constants of air ................................................................................................... 4
B.3 General relation for field admittance and intensity ........................................................... 7
B.4 Integral relations ................................................................................................................ 8
B.S Green's functions and formalism ...................................................................................... 9
B.6 Orthogonality of modes in a duct with locally reacting walls ......................................... 17
B.7 Orthogonality of modes in a duct with laterally reacting walls ....................................... 18
B.8 Source conditions ............................................................................................................ 18
B.9 Sommerfeld's condition ................................................................................................... 20
B.IO Principles of superposition ............................................................................................. 20
B.ll Hamilton's principle ....................................................................................................... 23
B.12. Adjoined wave equation .................................................................................................. 24
B.13 Vector and tensor formulation offundamentals .............................................................. 25
B.14 Boundary condition at a moving boundary ..................................................................... 37
B.IS Boundary conditions at liquids and solids ...................................................................... 38
Formulas of Acoustics VIII Contents
B.16 Corner conditions ............................................................................................................ 39
B.17 Surface wave at locally reacting plane ............................................................................. 39
B.18 Surface wave along a locally reacting cylinder ................................................................. 41
B.19 Periodic structures, admittance grid ................................................................................ 42
B.20 Plane wall with wide grooves .......................................................................................... 47
8.21 Thin grid on half-infinite porous layer ............................................................................ 49
B.22 Grid of finite thickness with narrow slits on half-infinite porous layer ......................... 52
8.23 Grid of finite thickness with wide slits on half-infinite porous layer. ............................ 54
c. Equivalent Networks
C.l Fundamentals of equivalent networks ............................................................................. 59
C.2 Distributed network elements ......................................................................................... 65
C.3 Elements with constrictions ............................................................................................ 70
CA Superposition of multiple sources in a network ............................................................. 71
C.5 Chain circuit .................................................................................................................... 71
D. Reflection of Sound
D.l Plane wave reflection at a locally reacting plane ............................................................. 74
0.2 Plane wave reflection at an infinitely thick porous layer ................................................ 75
D.3 Plane wave reflection at a porous layer of finite thickness ............................................. 76
DA Plane wave reflection at a multiple layer absorber. ......................................................... 78
D.5 Diffuse sound reflection at a locally reacting plane ......................................................... 79
0.6 Diffuse sound reflection at a bulk reacting porous layer ................................................. 81
D.7 Sound reflection and scattering at finite-size absorbers .................................................. 82
D.8 Uneven, local absorber surface ........................................................................................ 86
0.9 Scattering at the border of an absorbent half-plane ......................................................... 88
Formulas of Acoustics IX Contents
0.10 Absorbent strip in a hard baffle wall, with far field distribution .................................... 89
0.11 Absorbent strip in a hard baffle wall, as variational problem ......................................... 91
0.12 Absorbent strip in a hard baffle wall, with Mathieu functions ....................................... 94
0.13 Absorption of finite-size absorbers as a problem of radiation ........................................ 98
0.14 A monopole line source above an infinite, plane absorber;
integration method ........................................................................................................... 99
0.15 A monopole line source above an infinite, plane absorber;
with principle of superposition .................................................................................... 107
0.16 A monopole point source above a bulk reacting plane, exact forms ............................. 109
0.17 A monopole point source above a locally reacting plane, exact forms .......................... 112
0.18 A monopole point source above a locally reacting plane,
exact saddle point integration ........................................................................................ 114
0.19 A monopole point source above a locally reacting plane, approximations ................... 117
0.20 A monopole point source above a bulk reacting plane, approximations ....................... 124
E. Scattering of Sound
E.l Plane wave scattering at cylinders ................................................................................. 129
E.2 Plane wave scattering at cylinders and spheres ............................................................. 132
E.3 Multiple scattering at cylinders and spheres ................................................................ 142
E.4 Cylindrical wave scattering at cylinders ........................................................................ 143
E.5 Cylindrical or plane wave scattering at a comer surrounded by a cylinder ................... 145
E.6 Plane wave scattering at a hard screen ........................................................................... 152
E.7 Cylindrical or plane wave scattering at a screen with an elliptical cylinder atop .......... 153
E.8 Uniform scattering at screens and dams ........................................................................ 158
E.9 Scattering at a flat dam .................................................................................................. 167
E.lO Scattering at a semicircular absorbing dam on absorbing ground ................................... 169
E.ll Scattering in random media, generaL ............................................................................ 173
x
Formulas of Acoustics Contents
E.12 Function tables for monotype scattering ...................................................................... 180
E.13 Sound attenuation in a forest ......................................................................................... 184
E.l4 Mixed monotype scattering in random media ............................................................... 186
E.l5 Multiple triple-type scattering in random media .......................................................... 191
E. I 6 Plane wave scattering at elastic cylindrical shell ........................................................... 202
E.17 Plane wave backscattering by a liquid sphere ............................................................... 205
E.l8 Spherical wave scattering at a perfectly absorbing wedge ............................................. 206
E.l9 Impulsive spherical wave scattering at a hard wedge .................................................... 208
E.20 Spherical wave scattering at a hard screen ..................................................................... 210
F. Radiation of Sound
F.I Definition of radiation impedance and end corrections ................................................. 214
F.2 Some methods to evaluate the radiation impedance ...................................................... 216
F.3 Spherical radiators ......................................................................................................... 218
F.4 Cylindrical radiators ...................................................................................................... 222
F.5 Piston radiator on a sphere ............................................................................................ 224
F.6 Strip-shaped radiator on cylinder .................................................................................. 226
F.7 Plane piston radiators .................................................................................................... 227
F.8 Uniform end correction of plane piston radiators ......................................................... 236
F.9 Narrow strip-shaped, field excited radiator ................................................................... 236
F .10 Wide strip-shaped, field excited radiator ....................................................................... 238
F.II Wide rectangular, field excited radiator .......................................................................... 240
F .12 End corrections .............................................................................................................. 243
F .13 Piston radiating into a hard tube .................................................................................... 253
F.14 Oscillating mass of a fence in a hard tube ...................................................................... 253
F .15 A ring-shaped piston in a baffle wall ............................................................................ 254
