The equation of motion for small displacements $\mathbf{u}$ in a homogeneous, isotropic, elastic material is

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

where $\lambda$ and $\mu$ are the Lamé constants. Derive the conditions satisfied by the polarisation $\mathbf{P}$ and (real) vector slowness s of plane-wave solutions $\mathbf{u}=\mathbf{P} f(\mathbf{s} \cdot \mathbf{x}-t)$, where $f$ is an arbitrary scalar function. Describe the division of these waves into $P$-waves, $S H$-waves and $S V$-waves.

A plane harmonic $S V$-wave of the form

$$\mathbf{u}=\left(s_{3}, 0,-s_{1}\right) \exp \left[i \omega\left(s_{1} x_{1}+s_{3} x_{3}-t\right)\right]$$

travelling through homogeneous elastic material of $P$-wave speed $\alpha$ and $S$-wave speed $\beta$ is incident from $x_{3}<0$ on the boundary $x_{3}=0$ of rigid material in $x_{3}>0$ in which the displacement is identically zero.

Write down the form of the reflected wavefield in $x_{3}<0$. Calculate the amplitudes of the reflected waves in terms of the components of the slowness vectors.

Derive expressions for the components of the incident and reflected slowness vectors, in terms of the wavespeeds and the angle of incidence $\theta_{0}$. Hence show that there is no reflected $S V$-wave if

$$\sin ^{2} \theta_{0}=\frac{\beta^{2}}{\alpha^{2}+\beta^{2}}$$

Sketch the rays produced if the region $x_{3}>0$ is fluid instead of rigid.