 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-4/16c/b/iii/solution

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

Substitution into

\nabla^{2} ϕ = 0

gives

Z^{''} - [(\frac{π}{L})^{2} + (\frac{2 π}{L})^{2}] Z = 0.

k = \frac{π 5}{L} .

Then

Z = A e^{k z} + B e^{- k z}

. Since the fluid occupies

z < 0

, decay as

z \to - \infty

B = 0

. Absorbing

A

into

F

and choosing

Z (0) = 1

gives

Z (z) = e^{k z}, k = \frac{π 5}{L} .

This is the usual exponential depth dependence of a deep-water gravity wave.

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