  In this paper we examine necessary conditions for an inhomogeneity to be
non-scattering, or equivalently, by negation, sufficient conditions for it to
be scattering. These conditions are formulated in terms of the regularity of
the boundary of the inhomogeneity. We examine broad classes of incident waves
in both two and three dimensions. Our analysis is greatly influenced by the
analysis carried out by Williams [28] in order to establish that a domain,
which does not possess the Pompeiu Property, has a real analytic boundary. That
analysis, as well as ours, relies crucially on classical free boundary
regularity results due to Kinderlehrer and Nirenberg [18], and Caffarelli [6].
