 Solution

Scale time by ${\omega }^{-1}$, length by $k^{-1}$, velocity by $\omega /k$, pressure by $\mu \omega$, and streamfunction by $\omega /k^2$. Write $A=\alpha /k$, $\xi =x-t$, and drop stars from dimensionless coordinates. The surface is $y=ϵ\sin \xi$ and its material velocity in the sheet frame is $(0,-ϵ\cos \xi )$. The no-slip boundary condition is thereforeThe stationary matrix identifies a preferred frame. In the sheet frame it translates with $\mathcal{U}{\mathbf{e}}_x$, so matrix-relative Brinkman velocity gives the consistent equationsEquivalently one may keep the printed right-hand side $A^2\mathbf{u}$ after defining $\tilde{p}=p-A^2\mathcal{U}x$; then the far-field pressure has the gradient needed to balance that term. Using the printed equation with a uniform far-field flow and no such adjustment is inconsistent. The distinction first matters at second order, since the first-order swimming speed is zero. We solve the upper half-space; the lower half-space is its reflected counterpart.