 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-1/40c/a/solution

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

Use a velocity potential proportional to

e^{ik (x - V t) + ik β_{1} y}

, where

β_{1} = (V / c_{1})^{2} - 1

. The wall condition gives amplitude

- V ϵ / β_{1}

. With

p^{'} = - ρ_{1} \partial_{t} ϕ

⟨ I_{y} ⟩ = \frac{ρ _{1} k ^{2} V ^{3} ϵ ^{2}}{2 β _{1}} .

For

V < c_{1}

the vertical wave number is imaginary: the disturbance is evanescent and carries no mean vertical energy flux.

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