INTERNATIONAL 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 This work is suhject to copyright. AII rights are reserved, whether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of illustrations, hroadcasting, reproduction hy photocopying machine 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