= Finite-depth Rayleigh-Taylor dispersion relation
{title2=$\omega^2=[\gamma k^3-g(\rho_1-\rho_2)k]/[\rho_1\coth(kL_1)+\rho_2\coth(kL_2)]$}
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.
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