= Solution
Put $\kappa=|k|>0$. Profiles satisfying the rigid-wall conditions are
$$
\varphi_1=A_1\cosh[\kappa(y-h_1)]e^{i(kx-\omega t)},\qquad
\varphi_2=A_2\cosh[\kappa(y+h_2)]e^{i(kx-\omega t)}.
$$
The kinematic conditions at zero give
$$
A_1=\frac{i\omega\eta_0}{\kappa\sinh(\kappa h_1)},\qquad
A_2=-\frac{i\omega\eta_0}{\kappa\sinh(\kappa h_2)}.
$$
Consequently $\varphi_{1t}(0)$ has amplitude $\omega^2\eta_0\coth(\kappa h_1)/\kappa$, whereas $\varphi_{2t}(0)$ 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>:
$$
\boxed{\omega^2=\frac{g(\rho_2-\rho_1)|k|}
{\rho_1\coth(|k|h_1)+\rho_2\coth(|k|h_2)}.}
$$
It is positive because the heavier fluid is below. As $k\to0$, $\omega^2\sim g(\rho_2-\rho_1)k^2/(\rho_1/h_1+\rho_2/h_2)$, 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.
Back to article page
