State the equations that relate strain to displacement and stress to strain in a uniform, linear, isotropic elastic solid with Lamé moduli $\lambda$ and $\mu$. In the absence of body forces, the Cauchy momentum equation for the infinitesimal displacements $\mathbf{u}(\mathbf{x}, t)$ is

$$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=\boldsymbol{\nabla} \cdot \boldsymbol{\sigma}$$

where $\rho$ is the density and $\boldsymbol{\sigma}$ the stress tensor. Show that both the dilatation $\boldsymbol{\nabla} \cdot \mathbf{u}$ and the rotation $\nabla \wedge \mathbf{u}$ satisfy wave equations, and find the wave-speeds $c_{P}$ and $c_{S}$.

A plane harmonic $\mathrm{P}$-wave with wavevector $\mathbf{k}$ lying in the $(x, z)$ plane is incident from $z<0$ at an oblique angle on the planar interface $z=0$ between two elastic solids with different densities and elastic moduli. Show in a diagram the directions of all the reflected and transmitted waves, labelled with their polarisations, assuming that none of these waves are evanescent. State the boundary conditions on components of $\mathbf{u}$ and $\boldsymbol{\sigma}$ that would, in principle, determine the amplitudes.

Now consider a plane harmonic P-wave of unit amplitude incident with $\mathbf{k}=$ $k(\sin \theta, 0, \cos \theta)$ on the interface $z=0$ between two elastic (and inviscid) liquids with wave-speed $c_{P}$ and modulus $\lambda$ in $z<0$ and wave-speed $c_{P}^{\prime}$ and modulus $\lambda^{\prime}$ in $z>0$. Obtain solutions for the reflected and transmitted waves. Show that the amplitude of the reflected wave is zero if

$$\sin ^{2} \theta=\frac{Z^{\prime 2}-Z^{2}}{Z^{\prime 2}-\left(c_{P}^{\prime} Z / c_{P}\right)^{2}}$$

where $Z=\lambda / c_{P}$ and $Z^{\prime}=\lambda^{\prime} / c_{P}^{\prime}$