Sound waves whose geometry is determined by a single dimension, plane waves, obey the wave equation
∂ 2 u ∂ r 2 − 1 c 2 ⋅ ∂ 2 u ∂ t 2 = 0where c designates the speed of sound in the medium. The monochromatic solution for plane waves will be taken to be
u ( r , t ) = sin ( k r ± ω t )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
∂ 2 u ∂ r 2 + 1 r ⋅ ∂ u ∂ r − 1 c 2 ⋅ ∂ 2 u ∂ t 2 = 0The monochromatic solution for cylindrical sound waves will be taken to be
u ( r , t ) = sin ( k r ± ω t ) rHere 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
∂ 2 u ∂ r 2 + 2 r ⋅ ∂ u ∂ r − 1 c 2 ⋅ ∂ 2 u ∂ t 2 = 0The monochromatic solution for spherical sound waves will be taken to be
u ( r , t ) = sin ( k r ± ω t ) rHere 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.
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