Show that the equations governing isotropic linear elasticity have plane-wave solutions, identifying them as $\mathrm{P}, \mathrm{SV}$ or $\mathrm{SH}$ waves.

A semi-infinite elastic medium in $y<0$ (where $y$ is the vertical coordinate) with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ is overlaid by a layer of thickness $h$ (in $0<y<h$ ) of a second elastic medium with density $\rho^{\prime}$ and Lamé moduli $\lambda^{\prime}$ and $\mu^{\prime}$. The top surface at $y=h$ is free, i.e. the surface tractions vanish there. The speed of S-waves is lower in the layer, i.e. $c_{S}^{\prime}{ }^{2}=\mu^{\prime} / \rho^{\prime}<\mu / \rho=c_{S}^{2}$. For a time-harmonic SH-wave with horizontal wavenumber $k$ and frequency $\omega$, which oscillates in the slow top layer and decays exponentially into the fast semi-infinite medium, derive the dispersion relation for the apparent wave speed $c(k)=\omega / k$,

$$\tan \left(k h \sqrt{\frac{c^{2}}{c_{S}^{2}}-1}\right)=\frac{\mu \sqrt{1-\frac{c^{2}}{c_{S}^{2}}}}{\mu^{\prime} \sqrt{\frac{c^{2}}{c_{S}^{\prime}}-1}} .$$

Show graphically that there is always one root, and at least one higher mode if $\sqrt{c_{S}^{2} / c_{S}^{\prime 2}-1}>\pi / k h$.