 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-4/38c/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The displacement must have small relative amplitude and small slope,

\frac{∣ η ∣}{h} ≪ 1, ∣ η_{x} ∣ ≪ 1.

Write the potentials as perturbations of the base flow,

ϕ_{\pm} = Ux + φ_{\pm}, ϕ_{0} = φ_{0},

with

∣\nabla φ_{\pm} ∣/ U

and

∣\nabla φ_{0} ∣/ U

small. A linearization evaluates the conditions at the unperturbed lines

y = \pm h

and discards products of perturbations. At

y = h

this gives

φ_{+, y} = η_{t} + U η_{x}, φ_{0, y} = η_{t}, φ_{+, t} + U φ_{+, x} = φ_{0, t},

whereas at

y = - h

it gives

φ_{-, y} = - η_{t} - U η_{x}, φ_{0, y} = - η_{t}, φ_{-, t} + U φ_{-, x} = φ_{0, t} .

The perturbation potentials remain harmonic functions, and their gradients decay at the corresponding far fields.

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