Evaluation of the Helmholtz boundary integral equation and its normal and tangential derivatives in two dimensions

2007 ◽  
Vol 301 (3-5) ◽  
pp. 864-877 ◽  
Author(s):  
S.A. Yang
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Two Dimensions ◽  
Boundary Integral

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

1980 ◽  
Vol 86 (1-2) ◽  
pp. 29-42 ◽  
Author(s):  
David Colton ◽  
Ralph Kleinman
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Boundary Integral Equation ◽  
Inverse Scattering ◽  
Dirichlet Boundary ◽  
Scattering Problem ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Moment Problems ◽  
Far Field

SynopsisThe 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.

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