Scattering by Penetrable Acoustic Targets

An acoustic target of constant density pt and variable index of refraction is imbedded in a surrounding acoustic fluid of constant density pa. A time harmonic wave propagating in the surrounding fluid is incident on the target. We consider two limiting cases of the target where the parameter e = pa/p, + 0 (the nearly rigid target) or E + ~0 (the nearly soft target). When the frequency of the incident wave is bounded away from the ‘in-vacua’ resonant frequencies of the target, the resulting scattered field is essentially the field scattered by the rigid target for E = 0 or the soft target if E + a). However, when the frequency of the incident wave is near a resonant frequency, the target oscillates and its interaction with the surrounding fluid produces peaks in the scattered field amplitude. In this paper we obtain asymptotic expansions of the solutions of the scattering problems for the nearly rigid and the nearly soft targets as E + 0 or E + co, respectively, that are uniformly valid in the incident frequency. The method of matched asymptotic expansions is used in the analysis. The outer and inner expansions correspond to the incident frequencies being far or near to the resonant frequencies, respectively. We have applied the method only to simple resonant frequencies, but it can be extended to multiple resonant frequencies. The method is applied to the incidence of a plane wave on a nearly rigid sphere of constant index of refraction. The far field expressions for the scattered fields, including the total scattering cross-sections, that are obtained from the asymptotic method and from the partial wave expansion of the solution are in close agreement for sufficiently small values of E.