Acoustic scattering by an impenetrable spheroid

The scattering of a plane acoustic wave from an impenetrable, soft or hard prolate or oblate spheroid is considered. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of spheroidal wave functions. In the second, a shape perturbation method, the field is expressed in terms of spheroidal wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. Analytical expressions are obtained for the scattered pressure field and the scattering cross-sections, when the solution is specialized to small values of h=d/(2a), (h << 1), with d the interfocal distance of the spheroid and 2a the length of its rotation axis. In this case exact, closed-form expressions, valid for each small h, are obtained for the expansion coefficients g (2) and g (4) in the relation S(h)=S(0)[1+g((2))h(2)+g((4))h(4)+O(h(6))] expressing the scattered field and the various scattering cross-sections.