We report on progress in Gutzwiller periodic orbit theory for elastodynamical resonators. The equations we solve are the conventional equations from continuum mechanics which are used to model phonon properties of mesoscopic structures as well as macroscopic bodies such as cars and bridges. The wave chaos is quantified in Gutzwiller theory which predicts the mode density using periodic classical orbits.
The complexity arises from the vectorial character of the elastic wave equation and the boundary conditions which for free boundaries leads to wave splitting and a multitude of surface waves. Nevertheless when these are incorporated a Gutzwiller description of the spectral density as a sum over periodic orbits arise . Formally from this the diagonal approximation can be made giving random matrix behaviour as observed also in experiments.
The talk will concentrate on solid bodies with simple shapes as for example discs and rectangles. Appropriate trace formulae for the density of modes differ here from the Gutzwiller trace formula in that sums over isolated orbits are replaced by sums over families of orbits.
We shall pay particular attention to orbits with incidence angles beyond the critical as these turn out to be less attenuated than ordinary ray splitting orbits. Furthermore surface orbits will be included as some for elastodynamics are virtually undamped at high frequencies. Similar surface orbits are also seen for impedance boundary condition in electrodynamics. Such behaviour does not occur in conventional scalar quantum mechanics where surface orbits are only important at low frequencies.
The ray splitting itself can be thought to introduce mixing in otherwise regular systems or more precisely a much larger number of periodic orbits arise. As a consequence even for simple shapes spectral statistics from integrable to chaotic can be observed contrary to scalar quantum mechanics.
 L. Couchmann, E. Ott and T.M. Antonsen, Jr., ``Quantum chaos in systems with ray splitting'', Phys. Rev. A 46, 6193 (1992).