Power-law near-shore ray asymptotics ($y-y_0\sim -K\sqrt{g\alpha }{x}^{1+p/2}/[\omega (1+p/2)]$)

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 ${\omega }^2\sim g{\kappa }^{2}h$, so $\kappa \sim \omega x^{-p/2}/\sqrt{g\alpha }$. The shoreward ray has $y^{\prime }\sim -K\sqrt{g\alpha }{x}^{p/2}/\omega$, 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.