(i) Sketch the rays in a small region near the relevant boundary produced by reflection and refraction of a $P$-wave incident (a) from the mantle on the core-mantle boundary, (b) from the outer core on the inner-core boundary, and (c) from the mantle on the Earth's surface. [In each case, the region should be sufficiently small that the boundary appears to be planar.]

Describe the ray paths denoted by $S S, P c P, S K S$ and $P K I K P$.

Sketch the travel-time $(T-\Delta)$ curves for $P$ and $P c P$ paths from a surface source.

(ii) From the surface of a flat Earth, an explosive source emits $P$-waves downwards into a stratified sequence of homogeneous horizontal elastic layers of thicknesses $h_{1}, h_{2}, h_{3}, \ldots$ and $P$-wave speeds $\alpha_{1}<\alpha_{2}<\alpha_{3}<\ldots$. A line of seismometers on the surface records the travel times of the various arrivals as a function of the distance $x$ from the source. Calculate the travel times, $T_{d}(x)$ and $T_{r}(x)$, of the direct wave and the wave that reflects exactly once at the bottom of layer 1 .

Show that the travel time for the head wave that refracts in layer $n$ is given by

$$T_{n}=\frac{x}{\alpha_{n}}+\sum_{i=1}^{n-1} \frac{2 h_{i}}{\alpha_{i}}\left(1-\frac{\alpha_{i}^{2}}{\alpha_{n}^{2}}\right)^{1 / 2}$$

Sketch the travel-time curves for $T_{r}, T_{d}$ and $T_{2}$ on a single diagram and show that $T_{2}$ is tangent to $T_{r}$.

Explain how the $\alpha_{i}$ and $h_{i}$ can be constructed from the travel times of first arrivals provided that each head wave is the first arrival for some range of $x$.