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 sca tterer which is either lossless or lossy. The point generated incident fiel d is normalized in such a way as to be able to recover the relative scatter ing solutions by plane wave excitation, as the location of the source appro aches infinity. For the case of a sphere, the low-frequency approximations of the zeroth, the first, and the second order are obtained in closed analy tic form for both, the lossy and the lossless case. The scattering amplitud e is obtained up to the third order The scattering, as well as the absorpti on cross-section are calculated up to the second order All results recover the case of plane wave incidence as the source recedes to infinity. Detaile d parametric analysis shows that if the point source is located approximate ly four radii away from the spherical scatterer, then the scattering charac teristics coincide with those generated from plane wave excitation. Further more, the dependence of the cross-sections on the ratio of the mass densiti es is analyzed. For the inverse scattering problem, we show that the second order approximation of the scattering cross-section is enough to obtain th e 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 place s. The inversion algorithm is stable as long as the locations of the excita tion points are not too far away from the scatterer. On the other hand if p hysical parameters are to be recovered from far field data, it seems that p lane wave excitation is more promising.