= Solution
The displacement must have small relative amplitude and small slope,
$$
\frac{|\eta|}{h}\ll1,\qquad |\eta_x|\ll1.
$$
Write the potentials as perturbations of the base flow,
$$
\phi_\pm=Ux+\varphi_\pm,\qquad \phi_0=\varphi_0,
$$
with $|\nabla\varphi_\pm|/U$ and $|\nabla\varphi_0|/U$ small. A <linearization> evaluates the conditions at the unperturbed lines $y=\pm h$ and discards products of perturbations. At $y=h$ this gives
$$
\varphi_{+,y}=\eta_t+U\eta_x,\qquad
\varphi_{0,y}=\eta_t,\qquad
\varphi_{+,t}+U\varphi_{+,x}=\varphi_{0,t},
$$
whereas at $y=-h$ it gives
$$
\varphi_{-,y}=-\eta_t-U\eta_x,\qquad
\varphi_{0,y}=-\eta_t,\qquad
\varphi_{-,t}+U\varphi_{-,x}=\varphi_{0,t}.
$$
The perturbation potentials remain <harmonic functions>, and their gradients decay at the corresponding far fields.
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