 Propulsive force of a periodic planar filament

For a small-amplitude planar filament without mean axial drift, resistive-force theory gives the leading time-averaged force on the filament as $F_x=({\xi }_{\perp }-{\xi }_{\parallel })T^{-1}{\int }_0^T{\int }_0^{\lambda }{y}_x{y}_{t}dxdt$. The parallel and perpendicular drag coefficients of a slender filament must differ to obtain propulsion at this order. A right-going wave gives negative axial force; a left-going wave gives positive axial force.