= Oscillatory bending of a moment-free semi-infinite filament
In units of <elastohydrodynamic penetration length> and inverse forcing frequency, $y_t=-y_{xxxx}$. Prescribing $y(0,t)=y_0\cos t$, zero bending moment $y_{xx}(0,t)=0$, and decay at infinity gives
$$
y(x,t)=\frac{y_0}{2}\left[e^{-Cx}\cos(t+Sx)+e^{-Sx}\cos(t-Cx)\right],\qquad C=\cos(\pi/8),\quad S=\sin(\pi/8).
$$
The two components have <phase velocities> $-1/S$ and $1/C$. Their unequal spatial attenuation produces a nonreciprocal shape cycle even though the endpoint actuation is reciprocal.
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