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This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in Rd for d = 2 or d = 3. Using ideas from transformation optics, we construct an approximate cloak by “blowing up” a small ball of radius ϵ > 0 to one of radius 1. In the anisotropic cloaking layer resulting from the “blow-up” change of variables, we incorporate a Drude-Lorentz- type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed ϵ) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as ϵ approaches 0, the L2 -norm of the scattered field outside the cloak, and its far field pattern, approach 0 uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves. 

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 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, and Caffarelli.