Finite-depth Rayleigh-Taylor dispersion relation (${\omega }^2=[\gamma k^3-g({\rho }_1-{\rho }_2)k]/[{\rho }_1\coth (k{L}_1)+{\rho }_2\coth (k{L}_2)]$)

For two resting incompressible inviscid layers between impermeable horizontal walls, the upper layer has density ${\rho }_1$ and depth $L_1$, and the lower layer has ${\rho }_2<{\rho }_1$ and depth $L_2$. Solving the Laplace equation for the velocity potential and imposing the linearized kinematic and capillary pressure conditions gives the displayed dispersion relation. With positive surface tension $\gamma$, exponential Rayleigh-Taylor instability occurs for $0<k<k_c$, where $k_c=\sqrt{g({\rho }_1-{\rho }_2)/\gamma }$. The depths change the growth rate, but not this cutoff.

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.