Scattering of a spherical wave by a small ellipsoid

A point generated incident field impinges upon a small triaxial ellipsoid w hich is arbitrarily oriented with respect to the point source. The point so urce field is so modified as to be able to recover the corresponding result s for plane wave incidence when the source recedes to infinity. The main di fficulty in solving analytically this low-frequency scattering problem conc erns the fitting of the spherical geometry, which characterizes the inciden t field, with the ellipsoidal geometry which is naturally adapted to the sc atterer. A series of techniques has been used which lead finally to analyti c solutions for the leading two low-frequency terms of the near as well as the far field. In contrast to the near-field approximations, which are expr essed in terms of ellipsoidal eigenexpansions, the far field is furnished b y a finite number of terms. This is very interesting because the constants entering the expressions of the Lame functions of degree higher than three are not obtainable analytically and therefore, in the near field, not even the Rayleigh approximation can be completely obtained. On the other hand, s ince only a few terms survive at the far field, the scattering amplitude an d the scattering cross-section are derived in closed form. It is shown that , in practice, if the source is located a distance equal to five or six tim es the biggest semiaxis of the ellipsoid the Rayleigh term of the approxima tion behaves almost as the incident field was a plane wave. The special cas es of spheroids, needles, discs, spheres as well as plane wave incidence ar e recovered. Finally, some theorems concerning monopole and dipole surface potentials are included.