= Solution
For the local phase $S$, define $k_i=\partial_iS$ and $\omega=-\partial_tS$. The <dispersion relation> is the <Hamilton-Jacobi equation> $S_t+\Omega(\nabla S,\mathbf x,t)=0$. Differentiate it in space and follow a ray with <group velocity> $\dot x_i=\Omega_{k_i}$; symmetry of the phase Hessian cancels the terms involving $\partial_i k_j$. Thus \b[the ray-tracing equations are]
$$
\boxed{\frac{dx_i}{dt}=\Omega_{k_i},\qquad\frac{dk_i}{dt}=-\Omega_{x_i},\qquad\frac{d\omega}{dt}=\Omega_t.}
$$
Here $d/dt=\partial_t+\dot{\mathbf x}\cdot\nabla$ means differentiation along the moving ray; subscripts on $\Omega$ mean partial derivatives holding its other independent arguments fixed.
The depth is stationary and independent of $y$, so $\omega=\omega_\infty$ and $l$ are constant. Choose the incident direction with positive $l$; the other sign reflects the picture. In the far deep water,
$$
\kappa_\infty=\omega_\infty^2/g,\qquad l=\kappa_\infty\sin\theta_\infty,\qquad k_\infty=-\kappa_\infty\cos\theta_\infty.
$$
Since $\Omega$ depends isotropically on the horizontal wavevector, the group-velocity ratio is $l/k$. Therefore \b[the ray equation is]
$$
\boxed{\frac{dy}{dx}=\frac lk,\qquad k=-\sqrt{\kappa(x)^2-l^2},\qquad \omega_\infty^2=g\kappa\tanh(\kappa\alpha x^p).}
$$
Near shore, $\kappa h\to0$, so the <shallow-water dispersion relation> gives $\kappa\sim\omega_\infty/(\sqrt{g\alpha}\,x^{p/2})$ and $k\sim-\kappa$. Integrating the ray slope yields
$$
\boxed{q=1+\frac p2,\qquad A=-\frac{l\sqrt{g\alpha}}{\omega_\infty(1+p/2)}=-\frac{\omega_\infty\sqrt{\alpha/g}\sin\theta_\infty}{1+p/2}.}
$$
For $p<2$, the near-shore phase is $S\sim ly-\omega_\infty t-\omega_\infty x^{1-p/2}/[\sqrt{g\alpha}(1-p/2)]$. A constant-phase <wave crest> thus has
$$
y-y_c\sim\frac{\omega_\infty}{l\sqrt{g\alpha}(1-p/2)}x^{1-p/2}.
$$
Its tangent becomes parallel to the shoreline, while rays approach perpendicular to it. The exponent is between zero and one, giving the curved crests sketched below for $p=1$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-3-shoreline-rays.png]
{title=Near-shore wavecrests bending parallel to the beach and rays approaching normally, for a depth proportional to x}
These are the formal <geometrical optics> asymptotics. For fixed $\alpha$ and $p<2$, $\kappa x\to0$ arbitrarily close to the shore, so the slowly varying wave approximation eventually fails there; the sketch describes the ray-theory regime before that final inner region.
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