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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.

The filament bending modulus has units force times length squared. The perpendicular drag coefficient in this equation is drag per unit length divided by velocity, so its units are force times time per length squared. Hence $A/(\zeta \omega )$ has units length to the fourth power. Balancing oscillatory viscous drag $\zeta \omega h$ against elastic force density $Ah/{\ell }^4$ gives the elastohydrodynamic penetration lengthIncreasing frequency confines the response nearer the actuator. The chosen coefficient one in this characteristic length is a convention; the exact decay lengths acquire the root-dependent factors found below.

The overdamped filament bending equation is linear, so sinusoidal forcing has a sinusoidal steady response at the same frequency. Write $h=h_0\mathrm{Re}[e^{i\omega t}F(\eta )]$ with $\eta =x/\ell$. Substitution givesUsing the elastohydrodynamic penetration length from the previous part reduces this ordinary differential equation toThe boundary conditions become $F(0)=1$, $F^{\prime \prime }(0)=0$, and decay of $F$ and its derivatives at infinity. The derivative condition expresses zero bending moment at the driven end, rather than a clamped slope. This separation of variables constructs the periodic response; an unspecified initial shape can add a transient solution of the homogeneous evolution problem.

Balancing the viscous and bending terms gives $\zeta \omega h\sim Ah/{\ell }^4$, so the elastohydrodynamic penetration length isInserting $h=h_0\mathrm{Re}[e^{i\omega t}F(x/\ell )]$ into the linear equation yields $F^{\prime \prime \prime \prime }=-iF$, with $F(0)=1$, $F^{\prime \prime }(0)=0$ and decay at infinity. Write $C=\cos (\pi /8)$, $S=\sin (\pi /8)$. Of the four roots of ${\lambda }^4=-i$, precisely two have negative real part:The two decaying solutions are $e^{{\lambda }_L\eta }$ and $e^{{\lambda }_R\eta }$, and the general decaying response is their linear combination. Both coefficients are nonzero: if one vanished, $F^{\prime \prime }(0)=0$ would force the other to vanish, contradicting $F(0)=1$.

Their real parts in the time-dependent response are proportional toConstant phases move respectively with velocities $-\omega \ell /S$ and $\omega \ell /C$, so these are damped travelling waves moving left and right. Since $S<C$, the right-going wave has the longer attenuation length and dominates at large $x$. Thus the less strongly damped, outward-moving wave dominates the far-field response. This is the oscillatory bending of a moment-free semi-infinite filament; the two boundary conditions determine its amplitudes without needing them explicitly here.

A periodic displacement with angular frequency $\omega$ has $y_t\sim \omega y$. Balancing this viscous forcing with bending over length ${\ell }_{\omega }$ gives $c_{\perp }\omega y\sim By/{\ell }_{\omega }^4$. Thus the elastohydrodynamic penetration length and sperm number areSince $[B/(c_{\perp }\omega )]={\mathsf{L}}^4$, the fourth root has dimensions length, as required.

With $X=x/{\ell }_{\omega }$, $\tau =\omega t$, and $Y=y/{\ell }_{\omega }$, the bending equation becomesThe defining balance cancels both coefficients, giving $\boxed{Y_{\tau }=-Y_{XXXX}}$. A large sperm number means that the periodically forced response decays before reaching the distant end.

The sperm number compares filament length with an elastohydrodynamic penetration length. For periodic forcing at angular frequency $\omega$, the usual convention isIts fourth power is a viscous-to-bending ratio on the full filament length. In steady cross-flow problems a different convention also occurs: $\mathrm{Sp}={\zeta }_{\perp }{L}^{3}U/A$, where $U$ is flow speed and $A$ the filament bending modulus. The defining formula distinguishes these conventions.