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A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half‐plane
Author(s) -
ChandlerWilde S.N.,
Peplow A.T.
Publication year - 2005
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200410157
Subject(s) - helmholtz equation , integral equation , mathematics , mathematical analysis , electric field integral equation , boundary value problem , mixed boundary condition , robin boundary condition , free boundary problem , poincaré–steklov operator , uniqueness theorem for poisson's equation
In this paper we study the application of boundary integral equation methods to the solution of the Helmholtz equation in a locally perturbed half‐plane with Robin or impedance boundary conditions. This problem models outdoor noise propagation from a cutting onto a surrounding flat plane, and also the harbour resonance problem in coastal engineering. We employ Green's theorem to derive a system of three coupled integral equations. The three unknowns are the pressure on the boundary of the disturbance and the pressure and its normal derivative on the interface with the upper half‐space. We prove that the integral equation formulation has a unique solution at all wavenumbers by proving equivalence of the boundary value problem and the integral equation formulation and proving uniqueness of solution for the boundary value problem.