Inverse scattering for an exterior Dirichlet problem
Author:
S. I. Hariharan
Journal:
Quart. Appl. Math. 40 (1982), 273-286
MSC:
Primary 78A45; Secondary 35J05
DOI:
https://doi.org/10.1090/qam/678198
MathSciNet review:
678198
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Abstract: In this paper we consider scattering due to a metallic cylinder which is in the field of a wire carrying a periodic current. The aim of the paper is to obtain information such as the location and shape of the cylinder with a knowledge of a far-field measurement in between the wire and the cylinder. The same analysis is applicable in acoustics in the situation that the cylinder is a soft-wall body and the wire is a line source. The associated direct problem in this situation is an exterior Dirichlet problem for the Helmholtz equation in two dimensions. We present an improved low-frequency estimate for the solution of this problem using integral equation methods, and our calculations on inverse scattering are accurate to this estimate. The far-field measurements are related to the solutions of boundary integral equations in the low-frequency situation. These solutions can be expressed in terms of mapping function which maps the exterior of the unknown curve onto the exterior of a unit disk. The coefficients of the Laurent expansion of the conformal transformations can be related to the far-field coefficients. The first far-field coefficient leads to the calculation of the distance between the source and the cylinder. The other coefficients are determined by placing the source in a different location and using the corresponding new far-field measurements.
D. L. Colton, The inverse scattering problem for a cylinder, Proc. Roy. Soc. Edinburgh A84, 135–143 (1979)
D. L. Colton and R. Kleinman, The direct and inverse scattering problems for an arbitrary cylinder : Dirichlet boundary conditions, Proc. Roy. Soc. Edinburgh A86, 29–42 (1980)
- S. I. Hariharan and R. C. MacCamy, Integral equation procedures for eddy current problems, J. Comput. Phys. 45 (1982), no. 1, 80–99. MR 650426, DOI https://doi.org/10.1016/0021-9991%2882%2990103-6
- George Hsiao and R. C. MacCamy, Solution of boundary value problems by integral equations of the first kind, SIAM Rev. 15 (1973), 687–705. MR 324242, DOI https://doi.org/10.1137/1015093
S. I. Hariharan and E. Stephan, A boundary element method in two-dimensional electromagnetics, ICASE Report No. 81-14, April 28, 1981
D. L. Colton, private communication
D. L. Colton, The inverse scattering problem for a cylinder, Proc. Roy. Soc. Edinburgh A84, 135–143 (1979)
D. L. Colton and R. Kleinman, The direct and inverse scattering problems for an arbitrary cylinder : Dirichlet boundary conditions, Proc. Roy. Soc. Edinburgh A86, 29–42 (1980)
S. I. Hariharan and R. C. MacCamy, Integral equation procedures for eddy current problems, J. Computational Phys. (to appear).
G. C. Hsiao and R. C. MacCamy, Solution of boundary value problems by integral equations of the first kind, SIAM Review 15, 687–705 (1973)
S. I. Hariharan and E. Stephan, A boundary element method in two-dimensional electromagnetics, ICASE Report No. 81-14, April 28, 1981
D. L. Colton, private communication
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Article copyright:
© Copyright 1982
American Mathematical Society