Derive the ray-tracing equations for the quantities $\mathrm{d} k_{i} / \mathrm{d} t, \mathrm{~d} \omega / \mathrm{d} t$ and $\mathrm{d} x_{i} / \mathrm{d} t$ during wave propagation through a slowly varying medium with local dispersion relation $\omega=\Omega(\mathbf{k}, \mathbf{x}, t)$, explaining the meaning of the notation $\mathrm{d} / \mathrm{d} t$.

The dispersion relation for water waves is $\Omega^{2}=g \kappa \tanh (\kappa h)$, where $h$ is the water depth, $\kappa^{2}=k^{2}+l^{2}$, and $k$ and $l$ are the components of $\mathbf{k}$ in the horizontal $x$ and $y$ directions. Water waves are incident from an ocean occupying $x>0,-\infty<y<\infty$ onto a beach at $x=0$. The undisturbed water depth is $h(x)=\alpha x^{p}$, where $\alpha, p$ are positive constants and $\alpha$ is sufficiently small that the depth can be assumed to be slowly varying. Far from the beach, the waves are planar with frequency $\omega_{\infty}$ and with crests making an acute angle $\theta_{\infty}$ with the shoreline.

Obtain a differential equation (with $k$ defined implicitly) for a ray $y=y(x)$ and show that near the shore the ray satisfies

$$y-y_{0} \sim A x^{q}$$

where $A$ and $q$ should be found. Sketch the shape of the wavecrests near the shoreline for the case $p<2$.