Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECTURES - No. 277
THEORETICAL ACOUSTICS
AND
NUMERICAL TECHNIQUES
EDITED BY
P. FILIPPI
LABORATOIRE DE MECANIQUE ET D'A COUSTIQUE
MARSEILLE
SPRINGER-VERLAG WIEN GMBH
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or similar means, and storage in data hanks.
© 1983 hy Springer-Verlag Wien
Originally published by Springer Verlag Wien-New York in 1983
ISBN 978-3-211-81786-5 ISBN 978-3-7091-4340-7 (eBook)
DOI 10.1007/978-3-7091-4340-7
CONTENTS
Page
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
Integral Equations in Acoustics
by P.J.T. Filippi ....... . 1
Finite Element Techniques for Acoustics
by M. Petyt .............. . 51
Wave Propagation above Layered Media
by D. Habault . . . . . . . . . . . . . . 105
Boundary Element Methods and their Asymptotic Convergence
by W.L. Wendland ....................... . 135
Boundary Value Problems Analysis and Pseudo-Differential Operators in Acoustics
by M. Durand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Parametrices, Singularities, and High Frequency Asymptotics in the Theory of
Sound Waves
by H.D. Alber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Solution Procedures for Interface Problems in Acoustics and Electromagnetics
by E. Stephan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
PREFACE
Though Acoustics is a part of continuous media mechanics,
the mathematical methods recently developed in solid mechGnics are almost ·
never used by acousticians. This is, of course, due to a lack of scienti
fic effort on noise and sound mathematical problems. But the main reason
is that the difficulties encountered with the wave equation are strongly
different from those which appear in solid mechanics. And the very conve
nient mathematical tools have been developed during the past fifteen years
only.
1/ ACOUSTICS AND CLASSICAL MATHEMATICS
Let, first, have a brief survey of the mathematical problems
appearing in Acoustics. The time-dependant governing equation is of hyer
bolice type (a much less simple case than the parabolic type ), and un
bounded domains must be considered as soon as environmental acoustics or
under-water propagation are concerned.
Because of the difficulty to solve the wave equation, and be
cause lots of noise and sound sources are periodic (or can be considered
as periodic ), the Helmholtz equation is more frequently used. If the pro
pagation domain is bounded, resonnances appear. If the propagation domain
is unbounded, the total energy involved is unbounded, too. For these rea
sons, the use of the classical variational techniques is much less easy
than for the heat equation, static solid mechanics, or incompressible
fluid dynamics. Another difficulty is that as soon as energy is lost within
the boundaries - and this generally the case - the operators involved are
not self-adjoint : consequently, the powerful spectral theory does not ap
ply in its classical form.
Another type of difficulty will appear if acoustic energy can
IV Preface
propagate in the boundaries, and this is often the case. Such boundaries
are known by acnuRticians as "non locally reacting surfaces " : their ef
fect is described by an integral relationship between the pressure and the
normal velocity. Such non-local boundary conditions cannot be accounted
for by the classical theories.
Nevertheless, during the first half of this century, lots of
progress have been made in Acoustical Engineering. And when the large com
puters appear, numerical techniques have been developed.
First of all, analytical solutions of particular diffraction
problems have been established. Room acoustics has been studied for simple
geometrical configurations. The basic tools are the spatial Fourier trans
form and the separation of variables. Then, for less simple geometrical
data, pertubation techniques have been used.
Another classical method is to use asymptotic expansions with
respect to the distance or the frequency, or other characteristic parame
ters. The Geometrical Theory of Diffraction belongs to this category ;
~oughthey are based on considerations which seem satisfactory to the phy
sicist, the results are not always proved.
In the last fifteen years, boundary integral equations have
been used in acoustic diffraction [and, simultaneously, in electromagne
tism ). More or less simple numerical procedures have been adopted, but
their convergence has been proved recently, only. At the same period, fi
nite elementsmethods have been succesfully used for solving problems in
bounded domains ; for unbounded domains, these methods appear to be less
efficient than the boundary integral equations method.
2/ THE MODERN MATHEMATICAL ANALYSIS
As far as the data [boundary surfaces, source distribution,
space characteristic parameters of the physical medium, ... )are described
by sufficiently regular functions, and if local boundary conditions are
considered, the existence and uniqueness theorems of the solution can be
proved, using very classical mathematics. But, in many practical cases,
the necessary· regularity hypothesis are not fulfilled : the boundaries can
have corners ; the actual sources are efficiently described by distribu-
Preface v
tions (think of multipole sources encountered in jet noise description J.
Non local boundary conditions are of practical interest : this is the case
of domains bounded by vibrating structures ; another example is provided
by under water sediments in which a sound wave can propagate. For harmo
nic time dependant problems, the best adapted theory is certainly that of
pseudo-differential operators and Poisson pseudo-kernels (existence and
uniqueness of the solution, eigenmodes, regularity theorems, wedge condi
tions, ... , are easily proved ), Furthermore, these recent mathematical
tools enables to prove the convergence of the numerical techniques used
to solve acoustic boundary integral equations. Moreover, it can be expec
ted that new boundary finite elements , well adapted to acoustic problems,
will be defined and proved to provide a faster convergence.
During the last ten years, the Fourier Integral Operators
theory has been developed to study hyperbolic partial differential equa
tions. The typical example is the wave equation. It is particularly effi
cient for wave-fronts propagation problems. Moreover, it provides high
frequency asymptotical results : as a consequence, some results of the
Geometrical Theory of Diffraction can now be justified, or proved to be
somewhat incorrect. It can be expected that, for the wave equation, the
Fourier Integral Operators theory will provide to the physicist as useful
results as those derived from the pseudo-differential operators theory.
3/ CONTENTS OF THE PRESENT COURSE
The linearisation of the fluid dynamics equations leading to
classical acoustics equations will be recalled briefly in the next section
of this preface. A short bibliography of the basic papers and books is
given at the end.
The first chapter is devoted to acoustic boundary integral
equations : it is first shown how to establish them. Then, a simple nume
rical technique is described and illustrated by various examples. An "ex
perimental" convergence is shown by comparing the numerical solution eJ
ther to analytical results, or to model experiments.
In the second chapter, the finite method is presen
el~~~nts
ted. Here again, the numerical technique is described from an engineering
VI Preface
point of view.
The third chapter deals with propagation problems above lay
ered media. It is an example of the use of the spatial Fourier transform.
The results presented are quite new. It is shown that various representa
tions of the solution can be obtained : the first two ones recieve a sim
ple physical interpretation, while the third one is well adapted to nume
rical computation. Moreover, the method here developed is very general and
can be applied to a wide class of problems of wave propagation above lay
erd media, or within a layered medium as well (think of waves in shallow
water bounded by stratified sediments J.
The convergence of the numerical solutions of boundary inte
gral equations is studied in the fourth chapter. Two main techniques are
presented and compared : it is shown that the collocation method is much
more efficient than the techniqhe based on the variational formulation of
the problem.Thisis satisfactory for the physicists who have mainly used
this simple procedure.
Chapter five is devoted to an introduction to Fourier Inte
gral Operators, Pseudo-Differential Operators, and Poisson Pseudo-Kernels.
The basis of these theories is presented in the scope of· acoustic problems
As an example, the boundary integral equation encountered in the diffrac
tion by a hard infinitely thin screen is established, and the so-called
wedge conditions are proved.
The sixth chapter illustrates the interest of the Fourier In
tegral Operators. A problem of singularities propagation and high frequen
cy asymptotics is solved.
The seventh and last chapters devoted to interface problems
in acoustics, and in electromagnetism as well. Though some of the results
are quite reasonable from a physical point of view, it is shown that their
mathematical proof cannot be established without the theories presented
in chapter five.
4/ THE EQUATIONS OF LINEAR ACOUSTICS
An acoustical motion is a pertubation of a fluid motion. So,
the different acoustic wave equations derive from the Navier-Stokes equa-
Preface VII
tions. The conservation equations of the mechanics of continuous media
can be established in two ways. One way is to describe the modifications
of an infinitely small part of the medium by exterior forces. The other
way is to express that various quantities (mass, momentum, energy J are
conserved within any arbitrary domain. The advantage of this last method
is that the necessary regularity assumptions on the unknown functions are
weaker.
4-1 The conservation equations :
Let S be an isolated physical medium : here, "isolated" means
that no mass sources exist. Let p(t,XJ be the volumic mass of the medium
at a point X, and timet. Consider, at time ~a volume n of S. If a motion
n
exists, the total mass M(t) of the particles contained in does not chan
n
ge (although the domain changes, of course J. Let U(t,Xl stand for the
particle velocity at the point X, and time t. The mass conservation rela
tionship
dM(t] ~t J
p(t, X) dv 0
dt
n
leads to :
(1) J{ap~ttXJ + div [ p(t,Xl U(t,XJ ]}dv = 0
n
From a mathematical point of view, this expression only requires that the
quantities involved are locally integrable. If they are defined everywhere,
since (1] must be true for any n, the mass conservation equation becomes :
(2] apa(t ,XJ + div [ p(t,XJ U(t,Xl ] 0 •
t
The second equation is the generalisation of the classical law of
dynamics, f=my, which relates the accelaration y of a point mass m to the
force f acting on it. In fact, if U is the velocity, an equivalent form is
f=d(mUJ/dt, which shows that the time variation of the momentum mU is
n.
equal to the exterior force acting on the point mass. For a domain the
VIII Preface
total momentum is expressed by
Q J p(t,Xl U(t,Xl dv .
Two kinds of forces can act on~ : volume forces, with density F~; and sur
face forces, with density L applied on a~, the boundary of ~. These sur
face forces are exerted by the particles of the domain (S-~l on ~. Physi
cal and mathematical considerations lead to express the vector L as the
product of a symetrical third-order tensor o and the outward unit vector
n normal to a~, i.e. :
L. o .. n. i=1,2,3
l lJ J
in cartesian co-ordinates. The tensor o is called the stress tensor. The
integral form of the momentum conservation equation is then :
(3) oij nj ds = J Fi dv i=1,2,3.
~
Using Ostrogradskhi theorem, and since equation (3) is valid for any do
main ~. one gets
a CpU. l
( 4) ~tl- + (pU.U.l . o ... F. i=1,2,3
a l J , J lJ 'J l
it is the differential form of the momentum conservation equation.
At the present step, it is useful to specify that the physi
cal system under consideration is a newtonian fluid. Experience has shown
that, for this kind of fluid, the stress tensor is given by :
6 .. +2l.JD ..
lJ lJ
Then a new variable appears : the pressure p. The parameters A and l.J are
the viscosity coefficients. We are now left with four scalar equations
