The linearised equation of motion governing small disturbances in a homogeneous elastic medium of density $\rho$ is

$$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+\mu) \nabla(\boldsymbol{\nabla} \cdot \mathbf{u})+\mu \nabla^{2} \mathbf{u}$$

where $\mathbf{u}(\mathbf{x}, t)$ is the displacement, and $\lambda$ and $\mu$ are the Lamé moduli.

(a) The medium occupies the region between a rigid plane boundary at $y=0$ and a free surface at $y=h$. Show that $S H$ waves can propagate in the $x$-direction within this region, and find the dispersion relation for such waves.

(b) For each mode, deduce the cutoff frequency, the phase velocity and the group velocity. Plot the latter two velocities as a function of wavenumber.

(c) Verify that in an average sense (to be made precise), the wave energy flux is equal to the wave energy density multiplied by the group velocity.

[You may assume that the elastic energy per unit volume is given by

$$\left.E_{p}=\frac{1}{2} \lambda e_{i i} e_{j j}+\mu e_{i j} e_{i j} \cdot\right]$$