A uniform elastic solid with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ occupies the region between rigid plane boundaries $y=0$ and $y=h$. Show that SH waves can propagate in the $x$ direction within this layer, and find the dispersion relation for such waves.

Deduce for each mode (a) the cutoff frequency, (b) the phase velocity, and (c) the group velocity.

Show also that for each mode the kinetic energy and elastic energy are equal in an average sense to be made precise.

[You may assume that the elastic energy per unit volume $W=\frac{1}{2}\left(\lambda e_{k k}^{2}+2 \mu e_{i j} e_{i j}\right)$.]