(i) From the surface of a flat Earth, an explosive source emits P-waves downward into a horizontal homogeneous elastic layer of uniform thickness $h$ and P-wave speed $\alpha_{1}$ overlying a lower layer of infinite depth and P-wave speed $\alpha_{2}$, where $\alpha_{2}>\alpha_{1}$. A line of seismometers on the surface records the travel time $t$ as a function of distance $x$ from the source for the various arrivals along different ray paths.

Sketch the ray paths associated with the direct, reflected and head waves arriving at a given position. Calculate the travel times $t(x)$ of the direct and reflected waves, and sketch the corresponding travel-time curves. Hence explain how to estimate $\alpha_{1}$ and $h$ from the recorded arrival times. Explain briefly why head waves are only observed beyond a minimum distance $x_{c}$ from the source and why they have a travel-time curve of the form $t=t_{c}+\left(x-x_{c}\right) / \alpha_{2}$ for $x>x_{c}$.

[You need not calculate $x_{c}$ or $t_{c}$.]

(ii) A plane $\mathrm{SH}$-wave in a homogeneous elastic solid has displacement proportional to $\exp [i(k x+m z-\omega t)]$. Express the slowness vector $\mathbf{s}$ in terms of the wavevector $\mathbf{k}=(k, 0, m)$ and $\omega$. Deduce an equation for $m$ in terms of $k, \omega$ and the S-wave speed $\beta$.

A homogeneous elastic layer of uniform thickness $h$, S-wave speed $\beta_{1}$ and shear modulus $\mu_{1}$ has a stress-free surface $z=0$ and overlies a lower layer of infinite depth, S-wave speed $\beta_{2}\left(>\beta_{1}\right)$ and shear modulus $\mu_{2}$. Find the vertical structure of Love waves with displacement proportional to $\exp [i(k x-\omega t)]$, and show that the horizontal phase speed $c$ obeys

$$\tan \left[\left(\frac{1}{\beta_{1}^{2}}-\frac{1}{c^{2}}\right)^{1 / 2} \omega h\right]=\frac{\mu_{2}}{\mu_{1}}\left(\frac{1 / c^{2}-1 / \beta_{2}^{2}}{1 / \beta_{1}^{2}-1 / c^{2}}\right)^{1 / 2}$$

By sketching both sides of the equation as a function of $c$ in $\beta_{1} \leqslant c \leqslant \beta_{2}$ show that at least one mode exists for every value of $\omega$.