 Solution

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

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

Write

A = g (ρ_{1} - ρ_{2})

and

ε = k_{c} L_{1} ≪ 1

. Throughout the unstable band,

k L_{1} \leq ε

and

coth (k L_{1}) = 1/ (k L_{1}) + O (k L_{1})

. Since

ρ_{2} < ρ_{1}

σ^{2} (k) = \frac{A k - γ k ^{3}}{ρ _{1} c o t h ( k L _{1} ) + ρ _{2}} = \frac{L _{1}}{ρ _{1}} (A k^{2} - γ k^{4}) [1 + O (ε)] .

The leading polynomial is maximized at

k^{2} = A / (2 γ)

, where it equals

L_{1} A^{2} / (4 ρ_{1} γ)

σ_{m} \sim \frac{g ( ρ _{1} - ρ _{2} )}{2} \frac{L _{1}}{ρ _{1} γ} .

This is a leading asymptotic expression; the exact maximum for a small but nonzero layer depth has corrections.

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