This chapter discusses an asymptotic analysis of acoustic scattering by nearly rigid or soft targets. The qualitative features of the field scattered by a penetrable acoustic target, insonified by a time periodic incident wave, depends on a number of factors. Among them are the frequency of the incident wave: the densities and indicies of refraction of the target (ρt and nt respectively) and the surrounding acoustic fluid (ρa and na, respectively), and the shape of the target. When the incident frequency is bounded away from all the “in vacuo” or resonant frequencies of the target, then the scattered field is given essentially by the field scattered either by the rigid or by the soft target, depending on whether ε is small or large.
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