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 givesThe 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.