A uniform elastic solid with wavespeeds $c_{P}$ and $c_{S}$ occupies the region $z<0$. An $S$-wave with displacement

$$\mathbf{u}=(\cos \theta, 0,-\sin \theta) \mathrm{e}^{\mathrm{i} k(x \sin \theta+z \cos \theta)-\mathrm{i} \omega t}$$

is incident from $z<0$ on a rigid boundary at $z=0$. Find the form and amplitudes of the reflected waves.

When is the reflected $P$-wave evanescent? Show that if the $P$-wave is evanescent then the amplitude of the reflected $S$-wave has the same magnitude as the incident wave, and interpret this result physically.