= Solution
Without <surface tension>, the exact <dynamic boundary condition for an inviscid interface> is equality of pressure, $p_1=p_2$, at $z=\zeta$. The <Unsteady Bernoulli equation> in each irrotational fluid is
$$
\partial_t\Phi_i+\tfrac12|\nabla\Phi_i|^2+p_i/\rho_i+gz=C_i(t).
$$
Choose the potential gauge so that $C_i=p_0/\rho_i+U_i^2/2$, its unperturbed value. Expanding the kinetic term as $U_i^2/2+U_i\partial_x\varphi_i$ and evaluating gravity at the displaced interface gives
$$
p_i|_{z=\zeta}=p_0-\rho_i[\partial_t\varphi_i+U_i\partial_x\varphi_i+g\zeta]
$$
to first order. Equality of these pressures gives
$$
\boxed{\rho_1[(\partial_t+U_1\partial_x)\varphi_1+g\zeta]=\rho_2[(\partial_t+U_2\partial_x)\varphi_2+g\zeta]\qquad(z=0).}
$$
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