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The inverse scattering problem for a cylinder†

Published online by Cambridge University Press:  14 November 2011

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

Let D be a bounded simply connected domain in the plane and Ω the unit disk. Let F(Θ;k) be the far field pattern arising from the scattering of an incoming plane wave by the obstacle D and let an(k) denote the nth Fourier coefficient of F. Then if f conformally maps ℝ2\D onto ℝ2\Ω, a “moment” problem is derived which expresses an(k) in terms of f−1 for small values of the wave number k. The solution of this moment problem then gives the Laurent coefficients of f−1 and hence ∂D.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 84 , Issue 1-2 , 1979 , pp. 135 - 143
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Ahner, J., Low frequency Neumann scattering problems in two dimensions. AFOSR Scientific ReportTR-72–1243. Univ. of Delaware, Newark, Delaware, 1972.Google Scholar
2Bergman, S. and Schiffer, M.. Kernel Functions and Elliptic Differential Equations in Mathematical Physics (New York: Academic Press, 1953).Google Scholar
3Bleistein, N. and Cohen, J. K.. A survey of recent progress on inverse problems. ONR Scientific Report MS-R-7806, Univ. of Denver, Denver, Colorado, 1977.Google Scholar
4Bojarski, N. N.. Inverse scattering. Company Report N00019–73-C-0312/F, prepared for Naval Air Systems Command, AD-775 235/5, 1974.Google Scholar
5Erdélyi, A. et al. Higher Transcendental Functions, Vol. II (New York: McGraw Hill, 1953).Google Scholar
6Garabedian, P.. An integral equation governing electromagnetic waves. Quart. App. Math. 12 (1955), 428–433.CrossRefGoogle Scholar
7Hill, R. N., Kleinman, R. and Pfaff, E. W.. Convergent long wavelength expansionmethod for two-dimensional scattering problems. Canad. J. Phys. 51 (1973), 1541–1564.CrossRefGoogle Scholar
8Hong, S. and Goodrich, R. F.. Application of conformal mapping to scattering and diffraction problems. In Electromagnetic Wave Theory, Pt II, 907–914, ed. Brown, J. (Oxford: Pergamon, 1967).CrossRefGoogle Scholar
9Kleinman, R. and Wendland, W.. On Neumann's method for the exterior Neumann problemfor the Helmholtz equation, J. Math. Anal. Appl. 57 (1977), 170–202.CrossRefGoogle Scholar
10Majda, A.. High frequency asymptotics for the scattering matrix and the inverse problem of acousticscattering. Comm. Pure Appl. Math. 30 (1977), 165–194.CrossRefGoogle Scholar
11Taylor, A. E.. Introduction to Functional Analysis (New York: Wiley, 1958).Google Scholar