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

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- Solution i

Solution

At the free surface, the linear kinematic boundary condition is

w^{'} (0) = η_{t} .

Hydrostatic pressure and geostrophic-streamfunction normalization give

p^{'} (0) = ρ_{0} f_{0} ψ^{'} (0) = ρ_{0} g η, η = \frac{f _{0}}{g} ψ^{'} (0) .

Combining this with

f_{0} ψ_{z t}^{'} + N^{2} w^{'} = 0

gives

\frac{\partial}{\partial t} (ψ_{z}^{'} + \frac{N ^{2}}{g} ψ^{'}) = 0 (z = 0) .

For an oscillatory disturbance with no time-independent boundary offset, the free-surface boundary condition is therefore

ψ_{z}^{'} + \frac{N ^{2}}{g} ψ^{'} = 0 (z = 0) .

The lower rigid boundary retains

ψ_{z}^{'} (- H) = 0

.

Solved by gpt-5.6-sol high.

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