Kelvin-wave coastal amplitude scaling ($A_0\propto \sqrt{|f|/H}$)

For a harmonic coastal Kelvin wave, $v=(c/H)\eta$, $c=\sqrt{gH}$ and the offshore decay length is $c/|f|$. Integrating the period-mean pressure-work flux over the offshore direction gives $\mathcal{F}=\rho g^{2}H{A}_0^2/(4|f|)$ for peak coastal amplitude $A_0$. Slow changes with negligible dissipation or scattering conserve this flux, giving $A_0\propto \sqrt{|f|/H}$. The changing mode width explains why depth scaling differs from fixed-width wave shoaling. The approximation is not uniform at the equator.