 Solution

Put $\kappa =|k|>0$. Profiles satisfying the rigid-wall conditions areThe kinematic conditions at zero giveConsequently ${\phi }_{1t}(0)$ has amplitude ${\omega }^2{\eta }_0\coth (\kappa h_1)/\kappa$, whereas ${\phi }_{2t}(0)$ has amplitude $-{\omega }^2{\eta }_0\coth (\kappa h_2)/\kappa$. Substituting into the dynamic condition yields the dispersion relation for interfacial gravity waves between rigid boundaries:It is positive because the heavier fluid is below. As $k\to 0$, ${\omega }^2\sim g({\rho }_2-{\rho }_1)k^2/({\rho }_1/h_1+{\rho }_2/h_2)$, while in the deep-layer limit it becomes $g({\rho }_2-{\rho }_1)|k|/({\rho }_1+{\rho }_2)$. The spatially uniform mode has zero restoring frequency.