= Clamped--clamped bending mode
{title2=$\cos\beta\cosh\beta=1$}
A <normal mode> of a uniformly bending <elastic filament> fixed in position and slope at both ends satisfies $W''''=k^4W$. With $\beta=kL>0$, the allowed <wavenumbers> solve $\cos\beta\cosh\beta=1$. The first root is $4.730040745$ and $\beta_n=(n+\tfrac12)\pi+O(e^{-(n+1/2)\pi})$ for $n\geq1$. The apparent root at zero is not an <eigenfunction>: the zero-eigenvalue cubic polynomial satisfying all four <clamped boundary conditions> is identically zero. This spectrum differs from that of a filament clamped only at one end.
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