 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-331/1/c/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

For

L_{2} \to \infty

ρ_{1} coth (k L_{1}) + ρ_{2} > 0

. The numerator is unchanged, so the cutoff is exactly independent of depth:

k_{c} = \frac{g ( ρ _{1} - ρ _{2} )}{γ} .

This remains exact before taking the thin-layer limit in the finite-depth Rayleigh-Taylor dispersion relation.

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