= Overdamped relaxation of a clamped--free filament
{title2=$\tau_1=\zeta_\perp L^4/(B\alpha_1^4)$}
For $\zeta_\perp h_t=-Bh_{xxxx}$, separated <normal modes> decay at rates $B\alpha_n^4/(\zeta_\perp L^4)$. Clamping at zero and vanishing moment and shear at $L$ give $\cos\alpha_n\cosh\alpha_n=-1$. A mode is
$$
\phi_n(s)=\cosh(\alpha_ns)-\cos(\alpha_ns)-\eta_n[\sinh(\alpha_ns)-\sin(\alpha_ns)],\qquad
\eta_n=\frac{\cosh\alpha_n+\cos\alpha_n}{\sinh\alpha_n+\sin\alpha_n},\quad s=x/L.
$$
The fourth-order operator is <self-adjoint>: two <integration by parts> operations give $\int f g''''=\int f''g''$ under these boundary conditions. Its zero eigenspace is trivial and its <eigenfunctions> form an orthogonal complete basis. Thus a nonzero projection on the first mode gives long-time decay proportional to $\phi_1(x/L)\exp[-B\alpha_1^4t/(\zeta_\perp L^4)]$, where $\alpha_1=1.8751040687\ldots$. If this projection vanishes, the lowest excited mode replaces it.
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