Acoustic wave equation

In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure p {displaystyle p} or particle velocity u as a function of position x and time t {displaystyle t} . A simplified form of the equation describes acoustic waves in only one spatial dimension, while a more general form describes waves in three dimensions. In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure p {displaystyle p} or particle velocity u as a function of position x and time t {displaystyle t} . A simplified form of the equation describes acoustic waves in only one spatial dimension, while a more general form describes waves in three dimensions. 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 wave equation describing sound in one dimension (position x {displaystyle x} ) is where p {displaystyle p} is the acoustic pressure (the local deviation from the ambient pressure), and where c {displaystyle c} is the speed of sound. Provided that the speed c {displaystyle c} is a constant, not dependent on frequency (the dispersionless case), then the most general solution is where f {displaystyle f} and g {displaystyle g} are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one ( f {displaystyle f} ) travelling up the x-axis and the other ( g {displaystyle g} ) down the x-axis at the speed c {displaystyle c} . The particular case of a sinusoidal wave travelling in one direction is obtained by choosing either f {displaystyle f} or g {displaystyle g} to be a sinusoid, and the other to be zero, giving where ω {displaystyle omega } is the angular frequency of the wave and k {displaystyle k} is its wave number.