This page displays an animation of a solution of
the wave
equation on the sphere (that is,
(∂²/∂t²−c²Δ)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 =
∑uℓ,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
(ΔY[ℓ,m] =
−ℓ(ℓ+1)·Y[ℓ,m]),
and uℓ,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 x|y|z-symmetric
means it is symmetric w.r.t. the plane where the given coordinate
vanishes (the plane z=0 being parallel to the view
plane), L²-conservative
means that the wave evolution conserves
the L² norm (∫φ²) throughout time (for each color channel
separately).
This version does not use WebGL. A different version using WebGL (faster, but less reliable) is also available.