Karp’s theorem in acoustic scattering theory
HTML articles powered by AMS MathViewer
- by David Colton and Andreas Kirsch
- Proc. Amer. Math. Soc. 103 (1988), 783-788
- DOI: https://doi.org/10.1090/S0002-9939-1988-0947658-0
- PDF | Request permission
Abstract:
Karp’s Theorem states that if the far field pattern corresponding to the scattering of a time harmonic plane acoustic wave by a sound-soft cylinder is of the form ${F_0}(k;\theta - \alpha )$ where $k$ is the wave number, $\theta$ the angle of observation and $\alpha$ the angle of incidence of the plane wave, then the cylinder must be circular. A new proof is given of this result and extended to the cases of scattering by a sound-hard obstacle and an inhomogeneous medium.References
- David L. Colton and Rainer Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 700400
- David Colton and Peter Monk, A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region, SIAM J. Appl. Math. 45 (1985), no. 6, 1039–1053. MR 813464, DOI 10.1137/0145064
- David Colton and Peter Monk, A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region. II, SIAM J. Appl. Math. 46 (1986), no. 3, 506–523. MR 841463, DOI 10.1137/0146034
- Philip J. Davis, Interpolation and approximation, Dover Publications, Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. MR 0380189
- A. J. Devaney, Acoustic tomography, Inverse problems of acoustic and elastic waves (Ithaca, N.Y., 1984) SIAM, Philadelphia, PA, 1984, pp. 250–273. MR 804623 A. Erdélyi et al., Higher transcendental Functions, vol. II, McGraw-Hill, New York, 1953. W. A. Imbriale and R. Mittra, The two dimensional inverse scattering problem, IEEE Trans. Antennas and Prop. AP-18 (1970), 633-642.
- Samuel N. Karp, Far field amplitudes and inverse diffraction theory, Electromagnetic waves, Univ. Wisconsin Press, Madison, Wis., 1962, pp. 291–300. MR 0129766
- A. Kirsch, Properties of far-field operators in acoustic scattering, Math. Methods Appl. Sci. 11 (1989), no. 6, 773–787. MR 1021400, DOI 10.1002/mma.1670110604
- A. G. Ramm, On completeness of the products of harmonic functions, Proc. Amer. Math. Soc. 98 (1986), no. 2, 253–256. MR 854028, DOI 10.1090/S0002-9939-1986-0854028-0
- B. D. Sleeman, The inverse problem of acoustic scattering, IMA J. Appl. Math. 29 (1980), no. 2, 113–142. MR 679225, DOI 10.1093/imamat/29.2.113
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 783-788
- MSC: Primary 35R30; Secondary 35J05, 35P25, 76Q05
- DOI: https://doi.org/10.1090/S0002-9939-1988-0947658-0
- MathSciNet review: 947658