Rayleigh scattering for the Kelvin-inverted ellipsoid

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**G. Dassios,*Scattering of acoustic waves by a coated pressure--Release ellipsoid*, J.A.S.A.**70**, 176-185 (1981)**[2]**G. Dassios,*On the harmonic radius and the capacity of an inverse ellipsoid*, J. Math. Phys.**27**, 835-836 (1988) MR**940346****[3]**G. Dassios and R. E. Kleinman,*On Kelvin inversion and low-frequency scattering*, SIAM Review**31**, 565-585 (1989) MR**1025482****[4]**G. Dassios and R. E. Kleinman,*On the capacity and Rayleigh scattering for a class of nonconvex bodies*, Quart. J. Mech. Appl. Math.**42**, 467-475 (1989) MR**1018522****[5]**E. W. Hobson,*The Theory of Spherical and Ellipsoidal Harmonics*, Chelsea, New York, 1955 MR**0064922****[6]**R. E. Kleinman and T. B. A. Senior,*Rayleigh Scattering*, in*Low and High Frequency Asymptotics*, V. K. Varadan and V. V. Varadan, Eds., North-Holland, Amsterdam, 1986 MR**901961****[7]**W. D. MacMilan,*The Theory of the Potential*, Dover, New York, 1958 MR**0100172****[8]**T. Miloh,*Forces and moments on a tri-axial ellipsoid in potential flow*, Israel J. Techn.**11**, 63-74 (1973) MR**0395470****[9]**T. Miloh,*The ultimate image singularities for external ellipsoidal harmonics*, SIAM Appl. Math.**26**, 334-344 (1973) MR**0337126****[10]**T. Miloh,*Maneuvering hydrodynamics of ellipsoidal forms*, J. Ship Res.**23**66-75 (1979)**[11]**B. D. Sleeman,*The low-frequency scalar Dirichlet scattering by a general ellipsoid*, IMA J. of Appl. Math.**3**, 291-312 (1967)**[12]**A. F. Stevenson,*Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer/wave length)*, J. Appl. Phys.**24**, 1134-1142 (1953) MR**0061552****[13]**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)**[14]**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)**[15]**V. Twersky,*Rayleigh Scattering*, Appl. Opt.**3**, 1150-1162 (1964)**[16]**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)

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
76Q05,
35J05,
78A45

Retrieve articles in all journals with MSC: 76Q05, 35J05, 78A45

Additional Information

DOI:
https://doi.org/10.1090/qam/1724304

Article copyright:
© Copyright 1999
American Mathematical Society