A semi-infinite elastic filament lies along the positive $x$-axis in a viscous fluid. When it is perturbed slightly to the shape $y=h(x, t)$, it evolves according to

$$\zeta h_{t}=-A h_{x x x x}$$

where $\zeta$ characterises the viscous drag and $A$ the bending stiffness. Motion is forced by boundary conditions

$$h=h_{0} \cos (\omega t) \quad \text { and } \quad h_{x x}=0 \quad \text { at } \quad x=0, \quad \text { while } \quad h \rightarrow 0 \quad \text { as } \quad x \rightarrow \infty$$

Use dimensional analysis to find the characteristic length $\ell(\omega)$ of the disturbance. Show that the steady oscillating solution takes the form

$$h(x, t)=h_{0} \operatorname{Re}\left[e^{i \omega t} F(\eta)\right] \quad \text { with } \quad \eta=x / \ell,$$

finding the ordinary differential equation for $F$.

Find two solutions for $F$ which decay as $x \rightarrow \infty$. Without solving explicitly for the amplitudes, show that $h(x, t)$ is the superposition of two travelling waves which decay with increasing $x$, one with crests moving to the left and one to the right. Which dominates?