Small disturbances in a homogeneous elastic solid with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ are governed by the equation

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

where $\mathbf{u}(\mathbf{x}, t)$ is the displacement. Show that a harmonic plane-wave solution

$$\mathbf{u}=\operatorname{Re}\left[\mathbf{A} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right]$$

must satisfy

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

where the wavespeeds $c_{P}$ and $c_{S}$ are to be identified. Describe mathematically how such plane-wave solutions can be classified into longitudinal $P$-waves and transverse $S V$ - and $S H$-waves (taking the $y$-direction as the vertical direction).

The half-space $y<0$ is filled with the elastic solid described above, while the slab $0<y<h$ is filled with a homogeneous elastic solid with Lamé moduli $\bar{\lambda}$ and $\bar{\mu}$, and wavespeeds $\bar{c}_{P}$ and $\bar{c}_{S}$. There is a rigid boundary at $y=h$. A harmonic plane $S H$-wave propagates from $y<0$ towards the interface $y=0$, with displacement

$$\operatorname{Re}\left[A e^{i(\ell x+m y-\omega t)}\right](0,0,1)$$

How are $\ell, m$ and $\omega$ related? The total displacement in $y<0$ is the sum of $(*)$ and that of the reflected $S H$-wave,

$$\operatorname{Re}\left[R A e^{i(\ell x-m y-\omega t)}\right](0,0,1)$$

Write down the form of the displacement in $0<y<h$, and determine the (complex) reflection coefficient $R$. Verify that $|R|=1$ regardless of the parameter values, and explain this physically.