= Travelling-wave stationarity of planar filament propulsion
{title2=$c=(\xi_\perp-\xi_\parallel)/(\Gamma\xi_\perp)$}
For the leading periodic <propulsive force of a periodic planar filament> and <power of a periodic planar filament> functionals, the <Euler-Lagrange equation> for $F_x+\Gamma(\dot W-\dot W_0)$ gives $(\xi_\perp-\xi_\parallel)y_{xt}+\Gamma\xi_\perp y_{tt}=0$. For nonzero thrust, $y_t$ obeys an advection equation and the active <travelling wave> is $f(x-ct)$, with $c=(\xi_\perp-\xi_\parallel)/(\Gamma\xi_\perp)$. A static periodic shape may be added without changing leading force or power. Stationarity is a necessary extremum condition, not by itself an existence theorem for a global maximum.
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