 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 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.