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

For $L_2\to \infty$, the denominator is ${\rho }_1\coth (k{L}_1)+{\rho }_2>0$. The numerator is unchanged, so the cutoff is exactly independent of depth:This remains exact before taking the thin-layer limit in the finite-depth Rayleigh-Taylor dispersion relation.