 Solution

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

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

For the growing normal mode in two deep layers,

σ^{2} (k) = \frac{A k - γ k ^{3}}{ρ _{1} + ρ _{2}}, A = g (ρ_{1} - ρ_{2}) .

It vanishes at the endpoints of the unstable band and has its unique interior maximum where

A - 3 γ k^{2} = 0

. Using

k_{c} = A / γ

gives

σ_{m} = [\frac{2 A k _{c}}{3 3 ( ρ _{1} + ρ _{2} )}]^{1/2} .

This is the maximum exponential growth rate, not the real oscillation frequency.

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