Waves propagating in a slowly-varying medium satisfy the local dispersion relation

$$\omega=\Omega(\mathbf{k}, \mathbf{x}, t)$$

in the standard notation. Give a brief derivation of the ray-tracing equations for such waves; a formal justification is not required.

An ocean occupies the region $x>0, \quad-\infty<y<\infty$. Water waves are incident on a beach near $x=0$. The undisturbed water depth is

$$h(x)=\alpha x^{p}$$

with $\alpha$ a small positive constant and $p$ positive. The local dispersion relation is

$$\Omega^{2}=g \kappa \tanh (\kappa h) \quad \text { where } \quad \kappa^{2}=k_{1}^{2}+k_{2}^{2}$$

and where $k_{1}, k_{2}$ are the wavenumber components in the $x, y$ directions. Far from the beach, the waves are planar with frequency $\omega_{\infty}$ and crests making an acute angle $\theta_{\infty}$ with the shoreline $x=0$. Obtain a differential equation (in implicit form) 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 appearance of the wavecrests near the shoreline.