Acoustic wave equation

Equation for the propagation of sound waves through a medium

In physics, the acoustic wave equation is a second order partial differential equation that governs the propagation of acoustic waves through a material medium resp. a standing wavefield. The equation describes the evolution of acoustic pressure p or particle velocity u as a function of position x and time t. A simplified (scalar) form of the equation describes acoustic waves in only one spatial dimension, while a more general form describes waves in three dimensions. Propagating waves in a pre defined direction can also be calculated using a first order one way wave equation. For lossy media, more intricate models need to be applied in order to take into account frequency dependent attenuation and phase speed. Such models include acoustic wave equations that incorporate fractional derivative terms, see also the acoustic attenuation article or the survey paper.

The following solutions are obtained by separation of variables in different coordinate systems. They are phasor solutions, that is they have an implicit time dependence factor of where is the angular frequency. The explicit time dependence is given by

Here is the wave number.

.

. Depending on the chosen Fourier convention, one of these represents an outward travelling wave and the other a nonphysical inward travelling wave. The inward travelling solution wave is only nonphysical because of the singularity that occurs at r=0; inward travelling waves do exist.