Integral Equations in Acoustics

• Paul J. T. Filippi
Chapter
Part of the International Centre for Mechanical Sciences book series (CISM, volume 277)

Abstract

The aim of this course is twofold. First, the integral equations of linear acoustics are established for both interior and exterior problems. The integral representation of the diffracted field has several advantages: a/ the regularity theorems of the solution are easily obtained using the theories of “pseudo-differential operators” [1, 2] and “Poisson pseudo-kernels”[3, 4], b/ it is probably the most convenient formulation when no-local boundary conditions are involved; c/ numerical methods provide analytical approximations of the total field which are very useful for exterior problems (far-field diffraction patterns are easily obtained, constant level curves can be drawn,...). Another significant result (which is not established here) concerns the so-called “edge-conditions” which appear when the propagation domain has a non-regular boundary, or more, when the diffracting obstacle is an infinitely thin screen. Such boundaries or obstacles can be considered as the limit of a sequence of regular boundaries or no-zero thickness regular obstacles. It can be shown that the corresponding sequence of solutions has an unique limit which belongs to a functional space, the properties of which depend on the boundary irregularities. The edge conditions are included in the definition of this functional space. The fundamental ideas of the modern symbolic calculus of the pseudo-differential operators theory were already described in the book “Multidimensional singular integral equations” by S.G. MIKHLIN [5]. But the method used by this author is rather complicated, and the proofs must be established for each particular case. The recent theories are of a great generality and the basic results, useful in acoustics, are very simple.

Keywords

Integral Equation Boundary Integral Equation Helmholtz Equation Sound Field Layer Potential

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