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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 39D / a / Solution

The linearized horizontal and vertical momentum, density, and incompressibility equations are

ρ_{0} u_{t} = - p_{x}, ρ_{0} w_{t} = - p_{z} - g ρ^{'}, ρ_{t}^{'} + w ρ_{0}^{'} = 0, u_{x} + w_{z} = 0.

Differentiate the momentum equations in time, take

\partial_{x}

of the vertical equation minus

\partial_{z}

of the horizontal one, and use incompressibility to eliminate

u, p,

and

ρ^{'}

. Neglecting the stated slow-background term gives

\nabla^{2} w_{tt} + N^{2} (z) w_{xx} = 0, N^{2} (z) = - \frac{g}{ρ _{0}} \frac{d ρ _{0}}{d z} .

Solved by gpt-5.6-sol high.

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