Direct and inverse scattering for point source fields. The penetrable small sphere

A point source field is disturbed by the presence of a small penetrable scatterer which is either lossless or lossy. The point generated incident field is normalized in such a way as to be able to recover the relative scattering solutions by plane wave excitation, as the location of the source approaches infinity. For the case of a sphere, the low-frequency approximations of the zeroth, the first, and the second order are obtained in closed analytic form for both, the lossy and the lossless case. The scattering amplitude is obtained up to the third order The scattering, as well as the absorption cross-section are calculated up to the second order All results recover the case of plane wave incidence as the source recedes to infinity. Detailed parametric analysis shows that if the point source is located approximately four radii away from the spherical scatterer, then the scattering characteristics coincide with those generated from plane wave excitation. Furthermore, the dependence of the cross-sections on the ratio of the mass densities is analyzed. For the inverse scattering problem, we show that the second order approximation of the scattering cross-section is enough to obtain the position, as well as the radius of an unknown sphere. This is achieved by considering the exciting point source to be located at five specific places. The inversion algorithm is stable as long as the locations of the excitation points are not too far away from the scatterer. On the other hand if physical parameters are to be recovered from far field data, it seems that plane wave excitation is more promising.