  The relation between the elastic wave equation for plane, isotropic bodies
and an underlying classical ray dynamics is investigated. We study in
particular the eigenfrequencies of an elastic disc with free boundaries and
their connection to periodic rays inside the circular domain. Even though the
problem is separable, wave mixing between the shear and pressure component of
the wave field at the boundary leads to an effective stochastic part in the ray
dynamics. This introduces phenomena typically associated with classical chaos
as for example an exponential increase in the number of periodic orbits.
Classically, the problem can be decomposed into an integrable part and a simple
binary Markov process. Similarly, the wave equation can in the high frequency
limit be mapped onto a quantum graph. Implications of this result for the level
statistics are discussed. Furthermore, a periodic trace formula is derived from
the scattering matrix based on the inside-outside duality between eigen-modes
and scattering solutions and periodic orbits are identified by Fourier
transforming the spectral density
