POINT-SOURCE EXCITATION IN DIRECT AND INVERSE SCATTERING - THE SOFT AND THE HARD SMALL SPHERE

A spherical wave emanating from a point source is scattered by either a soft or a hard body. The incident spherical wave has a wavelength wh ich is much larger than the characteristic dimension of the scatterer and it is modified in such a way as to recover the plane wave incidenc e when the source point recedes to infinity. Using low frequency expan sions the scattering problem is transformed to a sequence of exterior potential problems in the presence of a monopole singularity located a t the source of the incident wave field. Complete expansions for the s cattering amplitude are provided. The method is applied to the cases o f a soft and a hard sphere and the first three approximations for the near, as well as the far, field are evaluated. It is observed that eve ry one, after the first, low frequency approximation of the far field, involves one spherical multipole more than the corresponding approxim ation for the case of an incident plane wave. As the point singularity tends to infinity, the relative results recover all the known express ions for plane incidence. It is shown that for point excitation the Ra yleigh approximation of the scattering amplitude for a hard sphere is of the second order, in contrast to the case of plane excitation which is of the third order. Simple algorithms that specify the radius and the position of a soft and a hard sphere are proposed, which are based on the additional dependence of the scattering amplitude represented by the distance from the point source to the centre of the scatterer. The inversion algorithm is shown to be stable whenever the source poin t is not too far away from the target sphere. A simple way to decide w hether the sphere is a soft or a hard body is also provided.