Exact solutions for wave resonances in rectangular pyramidal cavity

The exact analytical solutions for wave resonances are obtained for the first time in a pyramidal cavity with a square base and orthogonal opposite side faces. The solutions are a sum of degenerate modes of a cube composed from six such pyramids. The pyramid modes with the wave vectors parallel and oblique to edges, face and space diagonals of the cube are considered. According to the Weyl's law, the cube modes can be used to calculate one fourth of all modes of the pyramid.