A layer of rock of shear modulus $\bar{\mu}$ and shear wave speed $\bar{c}_{s}$ occupies the region $0 \leqslant y \leqslant h$ with a free surface at $y=h$. A second rock having shear modulus $\mu$ and shear wave speed $c_{s}>\bar{c}_{s}$ occupies $y \leqslant 0$. Show that elastic $S H$ waves of wavenumber $k$ and phase speed $c$ can propagate in the layer with zero disturbance at $y=-\infty$ if $\bar{c}_{s}<c<c_{s}$ and $c$ satisfies the dispersion relation

$$\tan \left[k h \sqrt{c^{2} / \bar{c}_{s}^{2}-1}\right]=\frac{\mu}{\bar{\mu}} \frac{\sqrt{1-c^{2} / c_{s}^{2}}}{\sqrt{c^{2} / \bar{c}_{s}^{2}-1}} .$$

Show graphically, or otherwise, that this equation has at least one real solution for any value of $k h$, and determine the smallest value of $k h$ for which the equation has at least two real solutions.