Stationary sets for the wave equation in crystallographic domains

Let be a crystallographic group in generated by reflections and let be the fundamental domain of We characterize stationary sets for the wave equation in when the initial data is supported in the interior of The stationary sets are the sets of time-invariant zeros of nontrivial solutions that are identically zero at .

We show that, for these initial data, the -dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group , This part comes from a corresponding odd symmetry of the initial data.

In physical language, the result is that if the initial source is localized strictly inside of the crystalline , then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one.