= Solution
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> are
$$
\boxed{\ell_\omega=\left(\frac B{c_\perp\omega}\right)^{1/4},\qquad \operatorname{Sp}=\frac L{\ell_\omega}=L\left(\frac{c_\perp\omega}{B}\right)^{1/4}.}
$$
Since $[B/(c_\perp\omega)]=\mathsf L^4$, the fourth root has dimensions length, as required.
With $X=x/\ell_\omega$, $\tau=\omega t$, and $Y=y/\ell_\omega$, the bending equation becomes
$$
c_\perp\omega\ell_\omega Y_\tau=-\frac B{\ell_\omega^3}Y_{XXXX}.
$$
The 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.
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