(i) What is the polarisation $\mathbf{P}$ and slowness $\mathbf{s}$ of the time-harmonic plane elastic wave $\mathbf{u}=\mathbf{A} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)] ?$

Use the equation of motion for an isotropic homogenous elastic medium,

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

to show that $\mathbf{s} \cdot \mathbf{s}$ takes one of two values and obtain the corresponding conditions on $\mathbf{P}$. If $\mathbf{s}$ is complex show that $\operatorname{Re}(\mathbf{s}) \cdot \operatorname{Im}(\mathbf{s})=0$.

(ii) A homogeneous elastic layer of uniform thickness $h, S$-wave speed $\beta_{1}$ and shear modulus $\mu_{1}$ has a stress-free surface $z=0$ and overlies a lower layer of infinite depth, $S$-wave speed $\beta_{2}\left(>\beta_{1}\right)$ and shear modulus $\mu_{2}$. Show that the horizontal phase speed $c$ of trapped Love waves satisfies $\beta_{1}<c<\beta_{2}$. Show further that

$$\tan \left[\left(\frac{c^{2}}{\beta_{1}^{2}}-1\right)^{1 / 2} k h\right]=\frac{\mu_{2}}{\mu_{1}}\left(\frac{1-c^{2} / \beta_{2}^{2}}{c^{2} / \beta_{1}^{2}-1}\right)^{1 / 2}$$

where $k$ is the horizontal wavenumber.

Assuming that (1) can be solved to give $c(k)$, explain how to obtain the propagation speed of a pulse of Love waves with wavenumber $k$.