Short wavelength approximation of a boundary integral operator for homogeneous and isotropic elastic bodies

A short wavelength approximation of a boundary integral operator for two-dimensional isotropic and homogeneous elastic bodies is derived from first principles starting from the Navier-Cauchy equation. Trace formulas for elastodynamics are deduced connecting the eigenfrequency spectrum of an elastic body to the set of periodic rays where mode conversion enters as a dynamical feature.

Figure 1

(Color online) Coordinates on the boundary: (a) Position representation with path of length $L$ from ${\mathbf{r}}^{\prime }\left(\alpha \right)$ to $\mathbf{r}\left(\beta \right)$ (here for an initial pressure wave); (b) momentum representation with shear and pressure path starting with tangential momentum $p^{\prime }$ and ending with momentum $p$ on the boundary.