 Solution

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.