= Solution
Write the <velocity potentials> in the upper, middle and lower domains as $\phi_+$, $\phi_0$ and $\phi_-$. <Irrotational flow> and <incompressible flow> imply the <Laplace equation>
$$
\nabla^2\phi_j=0,\qquad j\in\{+,0,-\}.
$$
The <far-field boundary conditions> are
$$
\nabla\phi_+\longrightarrow U\mathbf e_x\quad(y\to+\infty),
\qquad
\nabla\phi_-\longrightarrow U\mathbf e_x\quad(y\to-\infty).
$$
Every shear layer is a material interface. The exact <kinematic boundary condition> at $y=h+\eta$ is
$$
\phi_{+,y}=\eta_t+\phi_{+,x}\eta_x,
\qquad
\phi_{0,y}=\eta_t+\phi_{0,x}\eta_x,
$$
and at $y=-h-\eta$ it is
$$
\phi_{-,y}=-\eta_t-\phi_{-,x}\eta_x,
\qquad
\phi_{0,y}=-\eta_t-\phi_{0,x}\eta_x.
$$
Here each equation is evaluated on the indicated moving interface.
Finally, <pressure continuity> is the <dynamic boundary condition for an inviscid interface>. The <Unsteady Bernoulli equation> gives
$$
\phi_{+,t}+\frac12|\nabla\phi_+|^2
=\phi_{0,t}+\frac12|\nabla\phi_0|^2
\quad(y=h+\eta),
$$
$$
\phi_{-,t}+\frac12|\nabla\phi_-|^2
=\phi_{0,t}+\frac12|\nabla\phi_0|^2
\quad(y=-h-\eta),
$$
after absorbing spatially constant functions of time into the potentials.
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