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We study the scattering of waves from a planar inclusion with constant refractive index, governed by the Helmholtz equation. It is well understood that singular inclusions—those whose boundary has a singular point—generically scatter every incident wave. Far less is known about the scattering behavior of regular inclusions. We consider a large class of regular inclusions and show that they generically scatter any (complex-analytic) incident wave. Focusing on incident plane waves, we prove that they always scatter from any regular inclusion whose refractive index is less than one. For inclusions with refractive index greater than one, under an additional convexity assumption, we obtain explicit wavenumber intervals in which scattering occurs. These intervals drift to infinity and expand, and are given by explicit formulas in terms of the geometry (directional widths) of the inclusion. If the inclusion is elongated (its maximal width is at least twice its minimal width), the union of these intervals covers the entire spectrum, and hence such inclusions scatter every incident plane wave. Our approach makes use of a connection between this scattering problem and the Schiffer/Pompeiu problem.

 

This is based on a joint work with Michael Vogelius.