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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.

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.