Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions

For a nonlocally perturbed half-space we consider the scattering of time-harmonic acoustic waves. A second kind boundary integral equation formulation is proposed for the sound-soft case, based on a standard ansatz as a combined single- and double-layer potential but replacing the usual fundamental...

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Bibliographic Details
Published in:SIAM journal on applied mathematics Vol. 66; no. 3; pp. 1002 - 1026
Main Authors: Chandler-Wilde, Simon N., Eric Heinemeyer, Potthast, Roland
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2006
Subjects:
Acoustics
Algorithms
Applied mathematics
Boundary conditions
Boundary value problems
Differential equations
Fourier transformations
Fourier transforms
Geometric planes
Helmholtz equations
Infinity
Integral equations
Mathematical functions
Mathematical surfaces
Methods
Multiplication & division
Propagation
Simulation
Uniqueness
ISSN:0036-1399, 1095-712X
Online Access:Get full text
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Summary:For a nonlocally perturbed half-space we consider the scattering of time-harmonic acoustic waves. A second kind boundary integral equation formulation is proposed for the sound-soft case, based on a standard ansatz as a combined single- and double-layer potential but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Due to the unboundedness of the surface, the integral operators are noncompact. In contrast to the two-dimensional case, the integral operators are also strongly singular, due to the slow decay at infinity of the fundamental solution of the three-dimensional Helmholtz equation. In the case when the surface is sufficiently smooth (Lyapunov) we show that the integral operators are nevertheless bounded as operators on$L^{2}(\Gamma)$and on$L^{2}(\Gamma) \cap BC(\Gamma)$and that the operators depend continuously in norm on the wave number and on Γ. We further show that for mild roughness, i.e., a surface Γ which does not differ too much from a plane, the boundary integral equation is uniquely solvable in the space$L^{2}(\Gamma) \cap BC(\Gamma)$and the scattering problem has a unique solution which satisfies a limiting absorption principle in the case of real wave number.
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ISSN:0036-1399
1095-712X
DOI:10.1137/050635262