A uniform elastic solid with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ occupies the region between rigid plane boundaries $z=0$ and $z=h$. Starting with the linear elastic wave equation, show that SH waves can propagate in the $x$-direction within this waveguide, and find the dispersion relation $\omega(k)$ for the various modes.

State the cut-off frequency for each mode. Find the corresponding phase velocity $c(k)$ and group velocity $c_{g}(k)$, and sketch these functions for $k, \omega>0$.

Define the time and cross-sectional average appropriate for a mode with frequency $\mathrm{~}$energy. [You may assume that the elastic energy per unit volume is $\frac{1}{2}\left(\lambda e_{k k}^{2}+2 \mu e_{i j} e_{i j}\right)$.]

An elastic displacement of the form $\mathbf{u}=(0, f(x, z), 0)$ is created in a region near $x=0$, and then released at $t=0$. Explain briefly how the amplitude of the resulting disturbance varies with time as $t \rightarrow \infty$ at the moving position $x=V t$ for each of the cases $0<V^{2}<\mu / \rho$ and $V^{2}>\mu / \rho$. [You may quote without proof any generic results from the method of stationary phase.]