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I am interested in the vibrations of a thin, flexible rod that would only be clamped at one point, properly I'd like to calculate its eigenvalue. But the way I learned it in wave mechanics doesn't seem to apply here. The equation is:

$$ \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}$$

with $u = u(x,t)$ defined has the micro-displacement in one transverse direction, $x$ the longitudinal direction of the string, $c$ the speed of sound. Or applying

$$u (0,t) = \partial_x u(x,t)|_{x=0}=0$$

has no non trivial solutions, and so no spectrum. My real interest is to calculate the vibrational spectrum of a cantilever clamped at one of it extremities, which also obeys a second degree wave equation. (and I know that software can calculate these spectra, with the same boundary conditions, using the same equations)

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If there is no tension in the string the wave velocity will be zero. Are you thinking of a string hanging from one end so the tension in the string is due to the weight of the string? –  John Rennie Feb 12 '13 at 18:44
I'm thinking of the tension has derived from the stiffness of the material, from its elasticity law $t_{ij}=C_{ijkl}\partial_ku_l$ with $C_{ijkl}$ the stiffness tensor, $t_{ij}$ the stress tensor. –  Learning is a mess Feb 12 '13 at 18:47
Rather than "string" you might say "flexible rod" or something similar, because to me (and evidently to @John) "string" denotes something with very little (approaching zero) resistance to bending. –  dmckee Feb 12 '13 at 19:14
Oh ok, well in this case, read rod, I didn't know about this implication of string (little background in mechanics...) –  Learning is a mess Feb 12 '13 at 22:15
Don't just say "well, read...", instead edit which will make your intent clearer to later visitors. Figuring out what edits would most improve a posts are the actual use case for which the comment facility was intended. –  dmckee Feb 12 '13 at 23:24
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1 Answer

up vote 3 down vote accepted

What you need is the Euler-Bernoulli beam theory. The last three pages of this PDF explain the eigenmodes.

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It would be great if the salient points from those links were summarized here. –  Chris White Feb 12 '13 at 23:34
Thanks for this information zonk, as it already been generalized to 3d (first $w(x,y)$ with $w$ the deflection along the z axis, the surface of the cantilever being parallel to the $x-y$ plane; and second to the full 3d $\vec{u}(x,y,z)$ as in my first formula, micro displacements being allowed in all directions)? –  Learning is a mess Feb 16 '13 at 14:22
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