Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 1 / c / ii / Solution

Write $A=g({\rho }_1-{\rho }_2)$ and $\epsilon =k_c{L}_1\ll 1$. Throughout the unstable band, $k{L}_1\le \epsilon$ and $\coth (k{L}_1)=1/(k{L}_1)+O(k{L}_1)$. Since ${\rho }_2<{\rho }_1$,The leading polynomial is maximized at $k^2=A/(2\gamma )$, where it equals $L_1{A}^2/(4{\rho }_1\gamma )$. Thus the thin-layer Rayleigh-Taylor growth asymptotics giveThis is a leading asymptotic expression; the exact maximum for a small but nonzero layer depth has corrections.