Finiteness Results for Non-Scattering Herglotz Waves: The case of inhomogeneities obtained by very general perturbations of disks

We study non-scattering phenomena associated with the time-harmonic Helmholtz equation in two dimensions. For very general classes of star-shaped domains, we show that there are at most finitely many wave numbers such that Herglotz incident waves with a fixed density function are non-scattering.