The equation of motion for small displacements $\mathbf{u}(\mathbf{x}, t)$ in a homogeneous, isotropic, elastic medium of density $\rho$ is

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

where $\lambda$ and $\mu$ are the Lamé constants. Show that the dilatation $\nabla \cdot \mathbf{u}$ and rotation $\nabla \wedge \mathbf{u}$ each satisfy wave equations, and determine the corresponding wave speeds $c_{P}$ and $c_{S}$.

Show also that a solution of the form $\mathbf{u}=\mathbf{A} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)]$ satisfies

$$\omega^{2} \mathbf{A}=c_{P}^{2} \mathbf{k}(\mathbf{k} \cdot \mathbf{A})-c_{S}^{2} \mathbf{k} \wedge(\mathbf{k} \wedge \mathbf{A})$$

Deduce the dispersion relation and the direction of polarization relative to $\mathbf{k}$ for plane harmonic $P$-waves and plane harmonic $S$-waves.

Now suppose the medium occupies the half-space $z \leqslant 0$ and that the boundary $z=0$ is stress free. Show that it is possible to find a self-sustained combination of evanescent $P$-waves and $S V$-waves (i.e. a Rayleigh wave), proportional to exp $[i k(x-c t)]$ and propagating along the boundary, provided the wavespeed $c$ satisfies

$$\left(2-\frac{c^{2}}{c_{S}^{2}}\right)^{2}=4\left(1-\frac{c^{2}}{c_{S}^{2}}\right)^{1 / 2}\left(1-\frac{c^{2}}{c_{P}^{2}}\right)^{1 / 2}$$

[You are not required to show that this equation has a solution.]