  We study a question arising in inverse scattering theory: given a penetrable
obstacle, does there exist an incident wave that does not scatter? We show that
every penetrable obstacle with real-analytic boundary admits such an incident
wave. At zero frequency, we use quadrature domains to show that there are also
obstacles with inward cusps having this property. In the converse direction,
under a nonvanishing condition for the incident wave, we show that there is a
dichotomy for boundary points of any penetrable obstacle having this property:
either the boundary is regular, or the complement of the obstacle has to be
very thin near the point. These facts are proved by invoking results from the
theory of free boundary problems.
