This page enables you to calculate the vibrational eigenmodes of a free homogeneous isotropic sphere. Two other pages can be used for multilayer spheres and matrix-embedded multilayer spheres.

*Disclaimer*: use at your own risk.

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) each time a parameter is changed or you
click for a different overtone.
The required parameters are the
longitudinal (v_{L}) and transverse (v_{T}) sound speeds
and the diameter of the sphere.
You also have to choose the symmetry of the vibration (spheroidal or
torsional) the angular momentum (ℓ) and the overtone index (n).

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

element for m=0.
Only the displacement of the points of the sphere in the x=0 plane are
represented. The displacement for the other points are obtained by
rotation around the vertical (z) axis.
The equilibrium position is represented by a filled circle.
For the spheroidal modes, the displacements are inside the plan of the
figure.
For the torsional modes, the displacement are perpendicular to the
figure. They are represented by arcs starting at the equilibrium
position and terminated by a disk at the displaced position. The colors
for the points moving before or behind the x=0 plan are different. The length
of the lines and the diameter of the disks are proportional to the
normalized displacement.

The displacements of the surface can be visualized in 3D for the 2ℓ+1 degenerate modes using one of the anisotropic calculators. The parameters are filled with the parameters above when following one of the following links:

- H. Lamb, Proceedings of the London Mathematical Society, s1-13: 189-212 (1881)
- A. C. Eringen and E. S. Suhubi,
*Elastodynamics*(Academic, New York, 1975), vol. II, pp. 804-833 ( isbn: 0122406028 )

(*Warning*: the calculation of the spheroidal ℓ=0 frequency is incorrect in this book.) - normalization of the displacement : D. B. Murray & L. Saviot, Phys. Rev. B 69, 094305 (2004)