= Solution
The interface is material. Applying the <kinematic boundary condition> to $y-\eta(x,t)=0$ gives, exactly,
$$
\phi_{-,y}=\eta_t+\phi_{-,x}\eta_x,\qquad
\phi_{+,y}=\eta_t+(U+\phi_{+,x})\eta_x
\quad(y=\eta).
$$
The disturbance <velocities> and $\eta$ are small. Products of two disturbance quantities are second order, and a <Taylor expansion> moves their evaluation to $y=0$ at <linear order>. Hence
$$
\boxed{\phi_{-,y}=\eta_t,\qquad
\phi_{+,y}=\eta_t+U\eta_x\quad(y=0).}
$$
The extra upper-fluid term is advection of the displaced interface by the base <uniform flow>.
Use the <Unsteady Bernoulli equation> separately on each side, with constant <mass density> $\rho$. There are no restoring terms specified at this internal interface, and its <pressure> is continuous. Choose the time-dependent potential gauges so that the undisturbed <pressure> is $p_0$ on both sides. Subtracting the base values from <Bernoulli's principle> gives, at <linear order>,
$$
p'_-=-\rho\phi_{-,t},\qquad
p'_+=-\rho(\phi_{+,t}+U\phi_{+,x}).
$$
The quadratic disturbance-<velocity> terms have been discarded, while the cross term with the base <velocity> is first order and must remain. Equating the perturbation <pressures> yields the third condition:
$$
\boxed{\phi_{-,t}=\phi_{+,t}+U\phi_{+,x}\quad(y=0).}
$$
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