three.js - multiple elements with text - webgl

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

$$\frac{{\partial}^{2}u}{\partial {r}^{2}}-\frac{1}{{c}^{2}}\cdot \frac{{\partial}^{2}u}{\partial {t}^{2}}=0$$where $c$ designates the speed of sound in the medium. The monochromatic solution for plane waves will be taken to be

$$u(r,t)=\mathrm{sin}(kr\pm \omega t)$$where $\omega $ is the frequency and $k=\omega /c$ 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

$$\frac{{\partial}^{2}u}{\partial {r}^{2}}+\frac{1}{r}\cdot \frac{\partial u}{\partial r}-\frac{1}{{c}^{2}}\cdot \frac{{\partial}^{2}u}{\partial {t}^{2}}=0$$The monochromatic solution for cylindrical sound waves will be taken to be

$$u(r,t)=\frac{\mathrm{sin}(kr\pm \omega t)}{\sqrt{r}}$$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

$$\frac{{\partial}^{2}u}{\partial {r}^{2}}+\frac{2}{r}\cdot \frac{\partial u}{\partial r}-\frac{1}{{c}^{2}}\cdot \frac{{\partial}^{2}u}{\partial {t}^{2}}=0$$The monochromatic solution for spherical sound waves will be taken to be

$$u(r,t)=\frac{\mathrm{sin}(kr\pm \omega t)}{r}$$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.