Semiclassical theory of flexural vibrations of plates

We study the biharmonic equation of flexural vibrations of elastic plates by a semiclassical method that can easily be generalized for other models of wave propagation. Three terms of the asymptotic number of levels for plates with smooth boundaries are derived and the trace formula for the density of states is obtained. The main difference between this formula and the Gutzwiller trace formula for billiards is the existence of a specific phase factor obtained while reflecting from the boundary. Six hundred eigenvalues of a clamped stadium plate are obtained by a specially developed numerical algorithm and the trace formula is assessed, looking at its Fourier transform. An extra contribution occurs for a free plate due to the existence of boundary modes.