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

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

where $\mathbf{u}(\mathbf{x}, t)$ is the displacement, and $\lambda$ and $\mu$ are the Lamé constants. Derive solutions for plane longitudinal waves $P$ with wavespeed $c_{P}$, and plane shear waves $S$ with wavespeed $c_{S}$.

The half-space $y<0$ is filled with the elastic solid described above, while the slab $0<y<h$ is filled with an elastic solid with shear modulus $\bar{\mu}$, and wavespeeds $\bar{c}_{P}$ and $\bar{c}_{S}$. There is a vacuum in $y>h$. A harmonic plane $S H$ wave of frequency $\omega$ and unit amplitude propagates from $y<0$ towards the interface $y=0$. The wavevector is in the $x y$-plane, and makes an angle $\theta$ with the $y$-axis. Derive the complex amplitude, $R$, of the reflected $S H$ wave in $y<0$. Evaluate $|R|$ for all possible values of $\bar{c}_{S} / c_{S}$, and explain your answer.