Theoretical Acoustics and Numerical Techniques

Abstract

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. Boundary Value Problems Analysis and Pseudo-Differential Operators in Acoustics 1/ ACOUSTICS AND CLASSICAL MATHEMATICSLet, first, have a brief survey of the mathematical problems appearing in Acoustics. The time-dependant governing equation is of hyerbolice type (a much less simple case than the parabolic type ), and unbounded 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 because lots of noise and sound sources are periodic (or can be considered as periodic ), the Helmholtz equation is more frequently used. If the propagation domain is bounded, resonnances appear. If the propagation domain is unbounded, the total energy involved is unbounded, too. For these reasons, 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 apply in its classical form.Another type of difficulty will appear if acoustic energy can 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 physicist, the results are not always proved.In the last fifteen years, boundary integral equations have been used in acoustic diffraction [and, simultaneously, in electromagnetism ). More or less simple numerical procedures have been adopted, but their convergence has been proved recently, only. At the same period, finite 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 ANALYSISAs 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 distribuPreface 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...