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The Quarterly Journal of Mechanics and Applied Mathematics 1997 50(2):321-332; doi:10.1093/qjmam/50.2.321
© 1997 by Oxford University Press
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THE IMPEDANCE SCATTERING PROBLEM FOR A POINT-SOURCE FIELD. THE SMALL RESISTIVE SPHERE

GEORGE DASSIOS and GREGORY KAMVYSSAS

( Division of Applied Mathematics, Department of Chemical Engineering, University of Patras Greece
Institute of Chemical Engineering and High Temperature Chemical Processes GR 265 000 Patras, Greece )

A small resistive scatterer disturbs a spherical time-harmonic field emanating from a point source. The incident point-source field is normalized in such a way as to be able to recover the corresponding results for plane-wave incidence. The full low-frequency expansion for the corresponding total field is reduced to an exterior boundary-value problem for the Laplace equation, which has to be solved repeatedly. Exact results for the case of a small resistive sphere are obtained. It is shown that the leading low-frequency approximations for the scattering as well as for the absorption cross-section are increasing functions of the impedance parameter and decreasing functions of the distance of the source from the scatterer. It is also shown that a small sphere scatters and absorbs more energy when it is illuminated by a point—rather than by a plane-wave field establishing the fact that the closer the source of illumination to the scatterer, the stronger the interaction. The leading approximation of.the absorption cross-section is independent of the wavenumber, while the leading approximation of the scattering cross-section is proportional to the second power of the wavenumber. Hence, in the low-frequency realm, absorption is by two orders of magnitude stronger than scattering. Finally, a comparison between point and plane-wave incidence, based on multipole expansions, is included.


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