Rayleigh scattering for the Kelvin-inverted ellipsoid

Dassios, George; Miloh, Touvia

doi:10.1090/qam/1724304

Authors: George Dassios and Touvia Miloh
Journal: Quart. Appl. Math. 57 (1999), 757-770
MSC: Primary 76Q05; Secondary 35J05, 78A45
DOI: https://doi.org/10.1090/qam/1724304
MathSciNet review: MR1724304
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Abstract: The Kelvin-inverted ellipsoid, with the center of inversion at the center of the ellipsoid, is a nonconvex biquadratic surface that is the image of a triaxial ellipsoid under the Kelvin mapping. It is the most general nonconvex 3-D body for which the Kelvin inversion method can be used to obtain analytic solutions for low-frequency scattering problems. We consider Rayleigh scattering by such a fourth-degree surface and provide all relevant analytical calculations possible within the theory of ellipsoidal harmonics. It is shown that only ellipsoidal harmonics of even degree are needed to express the capacity of the inverted ellipsoid. Special cases of prolate or oblate spheroids and that of the sphere are recovered through appropriate limiting processes. The crucial calculations of the norm integrals, which are expressible in terms of known ellipsoidal harmonics, are outlined in Appendix B.

G. Dassios, Scattering of acoustic waves by a coated pressure—Release ellipsoid, J.A.S.A. 70, 176–185 (1981)

George Dassios, On the harmonic radius and the capacity of an inverse ellipsoid, J. Math. Phys. 29 (1988), no. 4, 835–836. MR 940346, DOI https://doi.org/10.1063/1.527979
George Dassios and R. E. Kleinman, On Kelvin inversion and low-frequency scattering, SIAM Rev. 31 (1989), no. 4, 565–585. MR 1025482, DOI https://doi.org/10.1137/1031126
George Dassios and Ralph E. Kleinman, On the capacity and Rayleigh scattering for a class of nonconvex bodies, Quart. J. Mech. Appl. Math. 42 (1989), no. 3, 467–475. MR 1018522, DOI https://doi.org/10.1093/qjmam/42.3.467
E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922
V. K. Varadan and V. V. Varadan (eds.), Low and high frequency asymptotics, Mechanics and Mathematical Methods. A Series of Handbooks. Series III: Acoustic, Electromagnetic and Elastic Wave Scattering, vol. 2, North-Holland Publishing Co., Amsterdam, 1986. MR 901960
William Duncan MacMillan, The theory of the potential, MacMillan’s Theoretical Mechanics, Dover Publications, Inc., New York, 1958. MR 0100172
Touvia Miloh, Forces and moments on a tri-axial ellipsoid in potential flow, Israel J. Tech. 11 (1973), no. 1-2, 63–74; corrections, ibid. 11 (1973), no. 3, 166. MR 395470
Touvia Miloh, The ultimate image singularities for external ellipsoidal harmonics, SIAM J. Appl. Math. 26 (1974), 334–344. MR 337126, DOI https://doi.org/10.1137/0126031

T. Miloh, Maneuvering hydrodynamics of ellipsoidal forms, J. Ship Res. 23 66–75 (1979)

B. D. Sleeman, The low-frequency scalar Dirichlet scattering by a general ellipsoid, IMA J. of Appl. Math. 3, 291–312 (1967)

A. F. Stevenson, Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer)/wavelength, J. Appl. Phys. 24 (1953), 1134–1142. MR 61552

J. W. Strutt, Lord Rayleigh, On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders and on the passage of electric waves through a circular aperture in a conducting screen, Philos. Mag. 44, 28–52 (1897)

W. Thomson, Lord Kelvin, Papers on Electrostatics and Magnetism, MacMillan, London, 1882. First Published in J. Math. Pure Appl. 10, p. 364 (1845) and 12, p. 256 (1847)

V. Twersky, Rayleigh Scattering, Appl. Opt. 3, 1150–1162 (1964)

W. E. Williams, Some results for low-frequency Dirichlet scattering by arbitrary obstacles and their application to the particular case of the ellipsoid, IMA J. of Appl. Math. 7, 111–118 (1971)