(i) State the equations that relate strain to displacement and stress to strain in a linear, isotropic elastic solid.

In the absence of body forces, the Euler equation for infinitesimal deformations of a solid of density $\rho$ is

$$\rho \frac{\partial^{2} u_{i}}{\partial t^{2}}=\frac{\partial \sigma_{i j}}{\partial x_{j}}$$

Derive an equation for $\mathbf{u}(\mathbf{x}, t)$ in a linear, isotropic, homogeneous elastic solid. Hence show that both the dilatation $\theta=\boldsymbol{\nabla} \cdot \mathbf{u}$ and the rotation $\boldsymbol{\omega}=\nabla \wedge \mathbf{u}$ satisfy wave equations and find the corresponding wave speeds $\alpha$ and $\beta$.

(ii) The ray parameter $p=r \sin i / v$ is constant along seismic rays in a spherically symmetric Earth, where $v(r)$ is the relevant wave speed $(\alpha$ or $\beta)$ and $i(r)$ is the angle between the ray and the local radial direction.

Express $\tan i$ and sec $i$ in terms of $p$ and the variable $\eta(r)=r / v$. Hence show that the angular distance and travel time between a surface source and receiver, both at radius $R$, are given by

$$\Delta(p)=2 \int_{r_{m}}^{R} \frac{p}{r} \frac{d r}{\left(\eta^{2}-p^{2}\right)^{1 / 2}} \quad, \quad T(p)=2 \int_{r_{m}}^{R} \frac{\eta^{2}}{r} \frac{d r}{\left(\eta^{2}-p^{2}\right)^{1 / 2}}$$

where $r_{m}$ is the minimum radius attained by the ray. What is $\eta\left(r_{m}\right)$ ?

A simple Earth model has a solid mantle in $R / 2<r<R$ and a liquid core in $r<R / 2$. If $\alpha(r)=A / r$ in the mantle, where $A$ is a constant, find $\Delta(p)$ and $T(p)$ for $\mathrm{P}$-arrivals (direct paths lying entirely in the mantle), and show that

$$T=\frac{R^{2} \sin \Delta}{A}$$

[You may assume that $\int \frac{d u}{u \sqrt{u-1}}=2 \cos ^{-1}\left(\frac{1}{\sqrt{u}}\right)$.]

Sketch the $T-\Delta$ curves for $\mathrm{P}$ and $\mathrm{PcP}$ arrivals on the same diagram and explain briefly why they terminate at $\Delta=\cos ^{-1} \frac{1}{4}$.