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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-61/3/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 61 / 3 / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

Let

Σ_{0}

be the uniform background surface density,

u_{0} = - S x e_{y}

, and write

Σ = Σ_{0} + Σ^{'}

,

u = u_{0} + v

,

Φ_{d} = Φ_{d 0} + Φ_{d}^{'}

. The background sheet has no horizontal gravitational gradient; its vertical field is irrelevant to the stated planar equations. Put

D_{0} = \partial_{t} - S x \partial_{y}

. Keeping first-order terms gives

D_{0} Σ^{'} + Σ_{0} (\partial_{x} v_{x} + \partial_{y} v_{y}) = 0,

D_{0} v_{x} - 2Ω v_{y} = - \partial_{x} (c_{s}^{2} \frac{Σ ^{'}}{Σ _{0}} + Φ_{d, m}^{'}), D_{0} v_{y} + (2Ω - S) v_{x} = - \partial_{y} (c_{s}^{2} \frac{Σ ^{'}}{Σ _{0}} + Φ_{d, m}^{'}),

\nabla^{2} Φ_{d}^{'} = 4 π G Σ^{'} δ (z), Φ_{d, m}^{'} = Φ_{d}^{'} (x, y, 0, t) .

The

- S v_{x}

contribution comes from

v \cdot \nabla u_{0}

, while the Coriolis acceleration supplies the remaining coupling. The prescribed tidal potential has no perturbation, and its background force already balances the background Coriolis term. The isothermal pressure linearizes to

c_{s}^{2} \nabla Σ^{'} / Σ_{0}

.

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