Screened first-order transverse sheet flow (${\psi }_1=f(y)\sin (x-t)$)

With $s=\sqrt{1+A^2}$, a dimensionless decaying first-order transverse sheet has $f=(s{e}^{-y}-e^{-sy})/(s-1)$, $f(0)=1$, $f^{\prime }(0)=0$, and pressure $-s(s+1)e^{-y}\cos (x-t)$. The coincident-root limit is $(1+y)e^{-y}$, the ordinary transverse mode of a Taylor swimming sheet.