Derive the ray-tracing equations

$$\frac{d x_{i}}{d t}=\frac{\partial \Omega}{\partial k_{i}}, \quad \frac{d k_{i}}{d t}=-\frac{\partial \Omega}{\partial x_{i}}, \quad \frac{d \omega}{d t}=\frac{\partial \Omega}{\partial t}$$

for wave propagation through a slowly-varying medium with local dispersion relation $\omega=\Omega(\mathbf{k} ; \mathbf{x}, t)$, where $\omega$ and $\mathbf{k}$ are the frequency and wavevector respectively, $t$ is time and $\mathbf{x}=(x, y, z)$ are spatial coordinates. The meaning of the notation $d / d t$ should be carefully explained.

A slowly-varying medium has a dispersion relation $\Omega(\mathbf{k} ; \mathbf{x}, t)=k c(z)$, where $k=|\mathbf{k}|$. State and prove Snell's law relating the angle $\psi$ between a ray and the $z$-axis to $c$.

Consider the case of a medium with wavespeed $c=c_{0}\left(1+\beta^{2} z^{2}\right)$, where $\beta$ and $c_{0}$ are positive constants. Show that a ray that passes through the origin with wavevector $k(\cos \phi, 0, \sin \phi)$, remains in the region

$$|z| \leqslant z_{m} \equiv \frac{1}{\beta}\left[\frac{1}{|\cos \phi|}-1\right]^{1 / 2}$$

By considering an approximation to the equation for a ray in the region $\left|z_{m}-z\right| \ll \beta^{-1}$, or otherwise, determine the path of a ray near $z_{m}$, and hence sketch rays passing through the origin for a few sample values of $\phi$ in the range $0<\phi<\pi / 2$.