= Solution
Linearize about the flat interface at rest, $\eta=0$, $\varphi_j=0$. Both potentials still satisfy <Laplace equation>, now on the fixed strips $0<y<h_1$ and $-h_2<y<0$, with the same rigid-wall conditions. Discarding products of perturbations and evaluating the interface conditions at $y=0$ gives
$$
\boxed{\eta_t=\varphi_{1y}=\varphi_{2y},\qquad
\rho_1\varphi_{1t}-\rho_2\varphi_{2t}=g(\rho_2-\rho_1)\eta
\quad\text{at }y=0.}
$$
For a <normal mode> proportional to $e^{i(kx-\omega t)}$, the vertical profiles satisfy $\Phi_j''-k^2\Phi_j=0$, the interface velocity is $-i\omega\eta$, and the wall-normal derivatives vanish. Smallness of the amplitude relative to wavelength and layer depths justifies this <linearization>.
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