three.js - multiple elements with text - webgl

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

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

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

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

\[ u(r,t) = \frac{ \sin( k r \pm ω t ) }{ \sqrt{ r } } \]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

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

\[ u(r,t) = \frac{ \sin( k r \pm ω t ) }{ r } \]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.