= Solution
Let the rapidly varying phase be $\Theta$, so $\mathbf k=\nabla\Theta$ and $\omega=-\Theta_t$. The dispersion law is the Hamilton-Jacobi equation $\Theta_t+\Omega(\nabla\Theta,\mathbf x,t)=0$. Differentiating it and following a path with <group velocity> gives the <ray tracing> equations
$$
\boxed{\dot{\mathbf x}=\Omega_{\mathbf k},\qquad
\dot{\mathbf k}=-\Omega_{\mathbf x},\qquad\dot\omega=\Omega_t.}
$$
The final relation is obtained by the chain rule along that path; its two spatial terms cancel.
The medium is stationary and independent of $y$, so $\omega=\omega_\infty$ and $k_2=K$ are conserved. For deep water far from shore, $\kappa_\infty=\omega_\infty^2/g$. Choose the incident branch with $k_1<0$ and $K=(\omega_\infty^2/g)\sin\theta_\infty>0$; choosing the opposite alongshore direction changes its sign. Since the <dispersion relation> depends on direction only through $\kappa$, the group-velocity component ratio is $k_2/k_1$. Hence the required implicit ray equation is
$$
\boxed{\frac{dy}{dx}=-\frac K{\sqrt{\kappa(x)^2-K^2}},\qquad
\omega_\infty^2=g\kappa(x)\tanh(\kappa(x)\alpha x^p).}
$$
Near shore, $\kappa h\to0$ and $\kappa\sim\omega_\infty/\sqrt{g\alpha}\,x^{-p/2}$. Thus $y'\sim-K\sqrt{g\alpha}\,x^{p/2}/\omega_\infty$, whose integral gives
$$
\boxed{y-y_0\sim A x^q,\qquad q=1+p/2,\qquad
A=-\frac{K\sqrt{g\alpha}}{\omega_\infty(1+p/2)}.}
$$
The rays approach the shoreline normally. A crest is a level set of $\int k_1dx+Ky-\omega t$. Its slope is $dy/dx=-k_1/K\sim\omega_\infty x^{-p/2}/(K\sqrt{g\alpha})$, so crests become parallel to the shoreline. Their separation in the shore-normal direction is $2\pi/|k_1|\sim2\pi\sqrt{g\alpha}x^{p/2}/\omega_\infty$ and decreases to zero. These are formal ray asymptotics; for $p<2$, the slow-variation assumption eventually fails arbitrarily close to the shoreline. The sketch shows a representative shallow-water case $p=1$; it illustrates these limiting directions rather than assuming that value in the derivation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ii/paper-3-shore-crests.png]
{title=Near-shore wavecrests become parallel to the shoreline while incident rays approach normally}
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