 Longitudinal mode of a Taylor swimming sheet

For tangential displacement $ϵa\sin (k(x-t))$, the first-order decaying streamfunction is $-kay{e}^{-ky}\cos (k(x-t))$. It obeys tangential material velocity $-ka\cos (k(x-t))$ and zero normal velocity on the reference boundary. Its second-order mean swimming coefficient is $-k^2{a}^2/2$.