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

Table of contents
- Solution vii

Solution

Form the

x

-momentum equation plus

1/ c

times continuity, multiply by

e^{\mp f y / 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

(\frac{\partial}{\partial t} \pm \frac{1}{c} \frac{\partial}{\partial x}) \int_{0}^{L} e^{\mp f y / c} (u + \frac{ϕ}{c}) d y = 0.

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