A semi-infinite elastic medium with shear modulus $\mu_{1}$ and shear-wave speed $c_{1}$ lies in $y<0$. Above it there is a layer $0 \leq y \leqslant h$ of a second elastic medium with shear modulus $\mu_{2}$ and shear-wave speed $c_{2}\left(<c_{1}\right)$. The top boundary $y=h$ is stress-free. Consider a monochromatic shear wave propagating at speed $c$ with wavenumber $k$ in the $x$-direction and with displacements only in the $z$-direction.

Obtain the dispersion relation

$$\tan k h \theta=\frac{\mu_{1} c_{2}}{\mu_{2} c_{1}} \frac{1}{\theta}\left(\frac{c_{1}^{2}}{c_{2}^{2}}-1-\theta^{2}\right)^{1 / 2}, \quad \text { where } \quad \theta=\sqrt{\frac{c^{2}}{c_{2}^{2}}-1} .$$

Deduce that the modes have a cut-off frequency $\pi n c_{1} c_{2} / h \sqrt{c_{1}^{2}-c_{2}^{2}}$ where they propagate at speed $c=c_{1}$.