 Thin-layer Rayleigh-Taylor growth asymptotics

For an infinitely deep lower layer and $k_c{L}_1\ll 1$, write $A=g({\rho }_1-{\rho }_2)$. Uniformly over the unstable band, the finite-depth Rayleigh-Taylor dispersion relation gives ${\sigma }^2\sim (L_1/{\rho }_1)(A{k}^2-\gamma k^4)$. Its maximum occurs at $k_m\sim k_c/\sqrt{2}$, with the displayed growth rate. In two deep layers the maximum instead occurs at $k_c/\sqrt{3}$ and obeys ${\sigma }_m^2=2A{k}_c/[3\sqrt{3}({\rho }_1+{\rho }_2)]$. Thus confinement strongly reduces growth while leaving the cutoff unchanged. These thin-layer expressions are leading asymptotics, not exact finite-depth identities.