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The direct and inverse scattering problems for an arbitrary cylinder: Dirichlet boundary conditions†

Published online by Cambridge University Press:  14 November 2011

David Colton  and
Ralph Kleinman
David Colton
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware, U.S.A.
Ralph Kleinman
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware, U.S.A.
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Synopsis

The exterior Dirichlet problem for the Helmholtz equation in two dimensions is reduced to a boundary integral equation which is soluble by iteration. A standard application of Green's theorem leads to boundary integral equations which are not uniquely soluble because the operator has an eigenvalue. The present approach modifies the operator in such a way that the former eigenvalue is in the resolvent spectrum for low frequencies. The results are applied to the inverse scattering problem wherein the far field is known for a limited frequency range and one seeks the curve on which a plane wave is incident and a Dirichlet boundary condition is assumed. The first iterate in the solution of the boundary integral equation is used to obtain a sequence of moment problems relating the Fourier coefficients of the far field to the coefficients of the Laurent expansion of the conformai transformation which maps the exterior of a circle onto the exterior of the unknown curve. These moment problems are soluble in terms of the mapping radius which in turn may be determined from scattered far field data for an incident plane wave from a second direction.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 1-2 , 1980 , pp. 29 - 42
Copyright
Copyright © Royal Society of Edinburgh 1980

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