Cross-resistance of an asymmetric planar waveform ($\delta ={\Lambda }^{-1}\int x_s{y}_{s}ds$)

An asymmetric periodic planar filament can have nonzero cross-tangent average $\delta$ despite periodic vertical position. Its translational resistance per arclength period is $c_{\perp }\Lambda [I+(\rho -1)M]$, with $M_{xx}=\beta$, $M_{xy}=\delta$ and $M_{yy}=1-\beta$. Full force balance can therefore yield transverse translation, so an axial-only propulsion formula needs a symmetry or constraint assumption.