Form the $x$-momentum equation plus $1/c$ times continuity, multiply by $e^{\mp fy/c}$, and integrate across the channel. Integration by parts makes the $v_y$ term cancel the Coriolis term, and $v=0$ removes the endpoint contribution. The result is the pair of <Kelvin-wave channel invariants>
$$
\boxed{
\left(\frac\partial{\partial t}\pm\frac1c\frac\partial{\partial x}\right)
\int_0^Le^{\mp fy/c}\left(u+\frac\phi c\right)dy=0.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 1 / vii / Solution

{c}

Exponentially weighted cross-channel integrals of velocity and pressure isolate the two oppositely propagating boundary Kelvin waves. Their conserved characteristic amplitudes supply the two integral constraints missing from a channel geostrophic-adjustment problem.


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