= Power-law near-shore ray asymptotics
{title2=$y-y_0\sim -K\sqrt{g\alpha}x^{1+p/2}/[\omega(1+p/2)]$}
For depth $h=\alpha x^p$ with $p>0$, stationary <surface gravity waves> conserve frequency $\omega$ and alongshore wavenumber $K$. Near shore their dispersion becomes $\omega^2\sim g\kappa^2h$, so $\kappa\sim\omega x^{-p/2}/\sqrt{g\alpha}$. The shoreward ray has $y'\sim-K\sqrt{g\alpha}x^{p/2}/\omega$, giving the displayed integrated scaling. Rays approach normally, while equal-phase crests approach parallel to the shore and their shore-normal spacing shrinks as $x^{p/2}$. These are asymptotics within the ray approximation; for $p<2$, its slow-variation assumption eventually fails arbitrarily close to the endpoint.
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