This page displays an animation of a solution of
the wave
equation on the sphere (that is,
(∂²/∂`t`²−`c`²Δ)`φ`=0 where Δ is
the Laplacian,
and the displayed function `φ` is a 3-component vector giving
the RGB channels). The initial condition is somewhat
random. More precisely, the function is computed through its
decomposition
in spherical
harmonics `φ` =
∑`u`_{ℓ,m}(`t`)·`Y`[`ℓ`,`m`]
(for −`ℓ`≤`m`≤`ℓ`, and here `ℓ`
only ranges from 0 through 8 because so the animation can be performed
in real time), where `Y`[`ℓ`,`m`] are
L²-normalized eigenvectors of the Laplacian
(Δ`Y`[`ℓ`,`m`] =
−`ℓ`(`ℓ`+1)·`Y`[`ℓ`,`m`]),
and `u`_{ℓ,m}(`t`) is
a sinusoidal function of `t` with frequency
(`c`/2π)·√(`ℓ`(`ℓ`+1)), its phase and
amplitude being chosen at random. (Click on the rerandom

button to choose a different random configuration.)

A variety of choice settings for the initial conditions is
available, giving different symmetries: projective

means the
wave function `φ` is antipodally symmetric (symmetric
w.r.t. the center of the sphere), axially 2|3-symmetric

means
it is 2|3-symmetric w.r.t. the `z` axis (the
axis `x`=`y`=0, which is viewed head
on), plane

means it is symmetric w.r.t. the plane where the given coordinate
vanishes (the plane `x`|`y`|`z`-symmetric`z`=0 being parallel to the view
plane), L²-conservative

means that the wave evolution conserves
the L² norm (∫`φ`²) throughout time (for each color channel
separately).

This version uses WebGL. A different version not using WebGL (slower, but more reliable) is also available.