= Solution
The total <velocity fields> on the two sides are
$$
\mathbf u_+=(U,0)+\nabla\phi_+,\qquad \mathbf u_-=\nabla\phi_-.
$$
Each base flow is divergence-free. <Incompressibility> therefore makes both disturbance <velocity potentials> satisfy the <Laplace equation>:
$$
\boxed{\phi_{+,xx}+\phi_{+,yy}=0,\qquad
\phi_{-,xx}+\phi_{-,yy}=0.}
$$
At the fixed horizontal walls, impermeability requires zero vertical <velocity>:
$$
\boxed{\phi_{+,y}(x,h,t)=0,\qquad\phi_{-,y}(x,-h,t)=0.}
$$
The discontinuity in tangential base <velocity> at the interface is a <vortex sheet>. The assumption of <irrotational flow> is understood separately in each open fluid region; it cannot hold distributionally across that sheet when $U\ne0$.
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