 Finite-amplitude propulsion of an inextensible periodic filament

With drag ratio $\rho =c_{\parallel }/c_{\perp }$, squared-tangent average $\beta ={\Lambda }^{-1}\int x_s^{2}ds$, and axial wave speed $V$, resistive-force theory gives the displayed axial swimming ratio. Pure axial free swimming additionally requires vanishing cross-tangent average, as in reflection-symmetric waveforms; otherwise a transverse force or translation is present.