= Two-fluid interfacial gravity-shear dispersion relation
For two inviscid incompressible half-spaces, lower density/<velocity> $\rho_1,U_1$, upper density/<velocity> $\rho_2,U_2$, downward <gravity> $g$ and <surface tension> $\sigma$, a mode $e^{i(kx-\omega t)}$ with $k>0$ obeys
$$
(\omega-k\overline U)^2=\frac{gk(\rho_1-\rho_2)+\sigma k^3}{\rho_1+\rho_2}-\frac{\rho_1\rho_2}{(\rho_1+\rho_2)^2}k^2(U_1-U_2)^2,\qquad\overline U=\frac{\rho_1U_1+\rho_2U_2}{\rho_1+\rho_2}.
$$
A negative right-hand side produces exponential growth. The gravitational term recovers <Rayleigh-Taylor instability> for heavy-over-light ordering; the shear term can destabilize otherwise stable stratification. Finite layer thickness, <viscosity>, compressibility and <magnetic tension> lie outside this interface model.
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