= Solution
The <Kelvin-wave coastal amplitude scaling> follows from the flux. For a wave with peak coastal amplitude $A_0$, $\overline{v\eta}=(c/H)A_0^2e^{-2r/L_R}/2$. Integrating the supplied flux over the offshore coordinate gives its magnitude
$$
\mathcal F=\rho gH\frac cH\frac{A_0^2}{2}\frac{L_R}{2}
=\frac{\rho gc^2A_0^2}{4|f_0|}
=\boxed{\frac{\rho g^2H A_0^2}{4|f_0|}.}
$$
The direction is the alongshore propagation direction; the expression above is positive magnitude. For slowly varying depth, negligible dissipation/reflection and no other source or sink, the transported <energy> per time is conserved. At fixed latitude,
$$
\boxed{A_0\propto H^{-1/2},\qquad
\frac{A_{0,2}}{A_{0,1}}=\sqrt{\frac{H_1}{H_2}}.}
$$
The coastal amplitude increases on entering shallower <water>. This is stronger than the familiar fixed-width $H^{-1/4}$ shoaling law because the Kelvin mode's offshore width also contracts, $L_R\propto\sqrt H$. The comparison is a slowly varying trapped-mode result; abrupt topography can scatter or reflect <energy> and invalidate simple flux conservation for one outgoing mode. The amplitude panel in <past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2018/5/b/solution> illustrates the scaling.
Back to article page
