 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ib/paper-2/16c/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

z = 0

, the two free-surface conditions for a normal mode give

- i σ_{n} ϕ + g ζ = 0, - i σ_{n} ζ - \partial_{z} ϕ = 0.

Eliminating

ζ

yields

σ_{n}^{2} ϕ = g \partial_{z} ϕ .

For the vertical factor found in part (b),

\frac{\partial _{z} ϕ}{ϕ}_{z = 0} = k_{n} tanh (k_{n} h) .

σ_{n}^{2} = g k_{n} tanh (k_{n} h) = \frac{g}{h} Ψ (k_{n} h),

where

Ψ (x) = x tanh x .

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