= Finite-amplitude propulsion of an inextensible periodic filament
{title2=$U/V=(1-\rho)(1-\beta)/[1+\beta(\rho-1)]$}
With drag ratio $\rho=c_\parallel/c_\perp$, squared-tangent average $\beta=\Lambda^{-1}\int x_s^2ds$, 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.
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