= Thin-layer Rayleigh-Taylor growth asymptotics
{title2=$\sigma_m\sim\tfrac12g(\rho_1-\rho_2)\sqrt{L_1/(\rho_1\gamma)}$}
For an infinitely deep lower layer and $k_cL_1\ll1$, write $A=g(\rho_1-\rho_2)$. Uniformly over the unstable band, the <finite-depth Rayleigh-Taylor dispersion relation> gives $\sigma^2\sim(L_1/\rho_1)(Ak^2-\gamma k^4)$. Its maximum occurs at $k_m\sim k_c/\sqrt2$, with the displayed growth rate. In two deep layers the maximum instead occurs at $k_c/\sqrt3$ and obeys $\sigma_m^2=2Ak_c/[3\sqrt3(\rho_1+\rho_2)]$. Thus confinement strongly reduces growth while leaving the cutoff unchanged. These thin-layer expressions are leading asymptotics, not exact finite-depth identities.
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