three.js - multiple elements with text - webgl

Sound waves whose geometry is determined by a single dimension, plane waves, obey the wave equation

∂2u ∂r2 − 1 c2 ∂2u ∂t2 = 0

where c designates the speed of sound in the medium. The monochromatic solution for plane waves will be taken to be

u(r,t) = sin(kr ± ωt)

where ω is the frequency and k=ω/c is the wave number. The sign chosen in the argument determines the direction of movement of the waves.

Here is a plane wave moving on a three-dimensional lattice of atoms:

Here is a plane wave moving through a three-dimensional random distribution of molecules:

Sound waves whose geometry is determined by two dimensions, cylindrical waves, obey the wave equation

∂2u ∂r2 + 1 r ∂u ∂r − 1 c2 ∂2u ∂t2 = 0

The monochromatic solution for cylindrical sound waves will be taken to be

u(r,t) = sin(kr ± ωt) √r

Here is a cylindrical wave moving on a three-dimensional lattice of atoms:

Here is a cylindrical wave moving through a three-dimensional random distribution of molecules:

Sound waves whose geometry is determined by three dimensions, spherical waves, obey the wave equation

∂2u ∂r2 + 2 r ∂u ∂r − 1 c2 ∂2u ∂t2 = 0

The monochromatic solution for spherical sound waves will be taken to be

u(r,t) = sin(kr ± ωt) r

Here is a spherical wave moving on a three-dimensional lattice of atoms:

Here is a spherical wave moving through a three-dimensional random distribution of molecules:

The mathematical description of sound waves can be carried to higher dimensions, but one needs to wait for Four.js and its higher-dimensional successors to attempt visualizations.