Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 3 / b / ii

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- Solution ii

Solution

For

ψ^{'} = A (x, y, t) cos \frac{nπ z}{H},

the surface value is simply

ψ^{'} (0) = A

. The dynamic free-surface condition from part i gives

η (x, y, t) = \frac{f _{0}}{g} A (x, y, t) .

The cosine is the leading small-surface-displacement approximation to the exact quasi-geostrophic vertical mode; its derivative vanishes at both rigid-boundary locations.

Solved by gpt-5.6-sol high.

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