Variational formulations for scattering in a three-dimensional acoustic waveguide

Variational formulations for direct time‐harmonic scattering problems in a three‐dimensional waveguide are formulated and analyzed. We prove that the operators defined by the corresponding forms satisfy a Gårding inequality in adequately chosen spaces of test and trial functions and depend analytica...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2008-05, Vol.31 (7), p.821-847
Main Authors: Arens, Tilo, Gintides, Drossos, Lechleiter, Armin
Format: Article
Language:English
Subjects:
Calculus of variations and optimal control
elliptic equations
Exact sciences and technology
Mathematical analysis
Mathematics
Partial differential equations
reduced wave equation (Helmholtz)
Sciences and techniques of general use
variational methods for second order
wave scattering
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Summary:Variational formulations for direct time‐harmonic scattering problems in a three‐dimensional waveguide are formulated and analyzed. We prove that the operators defined by the corresponding forms satisfy a Gårding inequality in adequately chosen spaces of test and trial functions and depend analytically on the wavenumber except at the modal numbers of the waveguide. It is also shown that these operators are strictly coercive if the wavenumber is small enough. It follows that these scattering problems are uniquely solvable except possibly for an infinite series of exceptional values of the wavenumber with no finite accumulation point. Furthermore, two geometric conditions for an obstacle are given, under which uniqueness of solution always holds in the case of a Dirichlet problem. Copyright © 2007 John Wiley & Sons, Ltd.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.947