Scattering of a spherical wave by a small ellipsoid

Abstract

A point generated incident eld impinges upon a small triaxial ellipsoid which is arbitrarily oriented with respect to the point source. The point source eld is so modied as to be able to recover the corresponding results for plane wave incidence when the source recedes to innity. The main difculty in solving analytically this low-frequency scattering problem concerns the tting of the spherical geometry, which characterizes the incident eld, with the ellipsoidal geometry which is naturally adapted to the scatterer. A series of techniques has been used which lead nally to analytic solutions for the leading two low-frequency terms of the near as well as the far eld. In contrast to the near-eld approximations, which are expressed in terms of ellipsoidal eigenexpansions, the far eld is furnished by a nite number of terms. This is very interesting because the constants entering the expressions of the Lam´ e functions of degree higher than three are not obtainable analytically and therefore, in the near eld, not even the Rayleigh approximation can be completely obtained. On the other hand, since only a few terms survive at the far eld, the scattering amplitude and the scattering cross-section are derived in closed form. It is shown that, in practice, if the source is located a distance equal to ve or six times the biggest semiaxis of the ellipsoid the Rayleigh term of the approximation behaves almost as the incident eld was a plane wave. The special cases of spheroids, needles, discs, spheres as well as plane wave incidence are recovered. Finally, some theorems concerning monopole and dipole surface potentials are included.