= Solution
Use the standard ideal-fluid assumptions of <incompressible flow>, <inviscid flow>, gravity $-g\mathbf e_y$, and no surface tension. <Irrotational flow> gives $\mathbf u_j=\nabla\varphi_j$ in each layer, and incompressibility gives <Laplace equation>
$$
\nabla^2\varphi_1=0\quad(\eta<y<h_1),\qquad
\nabla^2\varphi_2=0\quad(-h_2<y<\eta).
$$
The rigid-wall <boundary conditions> are $\varphi_{1y}=0$ at $y=h_1$ and $\varphi_{2y}=0$ at $y=-h_2$. At the moving interface, each fluid has the same normal velocity as the interface:
$$
\boxed{\eta_t+\varphi_{jx}\eta_x=\varphi_{jy}\quad\text{at }y=\eta,\quad j=1,2.}
$$
The dynamic condition is continuity of pressure, $p_1=p_2$ at $y=\eta$. <Bernoulli equation> in each fluid reads $\varphi_{jt}+\tfrac12|\nabla\varphi_j|^2+p_j/\rho_j+gy=C_j(t)$. Choosing the potential gauges to subtract a common reference pressure, the interface condition becomes
$$
\boxed{\rho_1\left(\varphi_{1t}+\tfrac12|\nabla\varphi_1|^2+g\eta\right)
=\rho_2\left(\varphi_{2t}+\tfrac12|\nabla\varphi_2|^2+g\eta\right).}
$$
Tangential velocities need not agree across an inviscid interface.
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