This page enables you to calculate the frequencies of the vibrations of a free homogeneous isotropic infinite cylinder.

*Disclaimer*: use at your own risk, in particular for
calculations with q≠0 which have *not* been thoroughly tested.

The results section is updated each time a parameter is modified.

Frequency:

Period:

Wavenumber:

Reduced frequencies:

- ξ = ω.R/v
_{L}= - η = ω.R/v
_{T}=

The calculation is carried out by your web browser (javascript).
The required parameters are the longitudinal (v_{L}) and
transverse (v_{T}) sound speeds and the diameter of the
cylinder.
You also have to choose the angular momentum (m), the overtone
index (n) and the wavevector along z (q).

The displacement corresponding to the vibration is also displayed
using the `<canvas>`

element.
Only the displacement of the points of the cylinder in the z=0 plane are
represented.
The equilibrium position is represented by a black dashed circle.
The displacement along z are represented varying the opacity (or
transparency) of the lines.
For q=0, the motion of all the points is either in the z=0 plane or
along z.
For m≠0, the degeneracy of the mode is 2 and the second mode is
obtained by rotating by π/2m.

The displacements can be visualized in 3D using one of the anisotropic calculators. The parameters are filled with the parameters above when following one of the following links:

If you use this calculator, you might be interested in the following papers: L. Saviot, Phys. Rev. B 97, 155420 (2018) and H. Portalès et al., ACS Nano 14, 4395 (2020).

- A. C. Eringen and E. S. Suhubi,
*Elastodynamics*(Academic, New York, 1975), vol. II, pp. 772-778 ( isbn: 0122406028 ) - Complex numbers and Bessel functions in javascript.
- Thank you Diego for your reminder regarding the kernel notion in linear algebra!