Spherical Waves

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 = ∑u,m(tY[,m] (for −m, and here only ranges from 0 through 8 because so the animation can be performed in real time), where Y[,m] are L²-normalized eigenvectors of the Laplacian (ΔY[,m] = −(+1)·Y[,m]), and u,m(t) is a sinusoidal function of t with frequency (c/2π)·√((+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 uses WebGL. A different version not using WebGL (slower, but more reliable) is also available.