Show that the equations governing linear elasticity have plane-wave solutions, distinguishing between $\mathrm{P}, \mathrm{SV}$ and $\mathrm{SH}$ 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, that is, the surface tractions vanish there. The speed of the S-waves is lower in the layer, that is, $c_{S}^{\prime 2}=\mu^{\prime} / \rho^{\prime}<\mu / \rho=c_{S}^{2}$. 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 horizontal wave speed $c(k)=\omega / k$ :

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

Show graphically that for a given value of $k$ there is always at least one real value of $c$ which satisfies equation $(*)$. Show further that there are one or more higher modes if $\sqrt{c_{S}^{2} / c_{S}^{\prime 2}-1}>\pi / k h .$