Transverse mode of a Taylor swimming sheet (${\psi }_1=b(1+ky)e^{-ky}\sin (k(x-t))$)

For normal displacement $ϵb\sin (k(x-t))$, the first-order decaying streamfunction is $b(1+ky)e^{-ky}\sin (k(x-t))$. It obeys zero tangential material velocity and normal velocity $-kb\cos (k(x-t))$ on the reference boundary. Its second-order mean swimming coefficient in the convention of far-field velocity $U{\mathbf{e}}_x$ is $k^2{b}^2/2$.