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.

**[1]**D. L. Colton,*The inverse scattering problem for a cylinder*, Proc. Roy. Soc. Edinburgh**A84**, 135-143 (1979)**[2]**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)**[3]**S. I. Hariharan and R. C. MacCamy,*Integral equation procedures for eddy current problems*, J. Computational Phys. (to appear). MR**650426****[4]**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) MR**0324242****[5]**S. I. Hariharan and E. Stephan,*A boundary element method in two-dimensional electromagnetics*, ICASE Report No. 81-14, April 28, 1981**[6]**D. L. Colton, private communication

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DOI:
https://doi.org/10.1090/qam/678198

Article copyright:
© Copyright 1982
American Mathematical Society