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

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