= Solution
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 length>
$$
\boxed{\ell(\omega)=\left(\frac{A}{\zeta\omega}\right)^{1/4},\qquad\omega>0.}
$$
Increasing 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.
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