Acoustic wave equation

In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium resp. a standing wavefield. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure or particle velocity u as a function of position x and time . 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 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 wave equation describing a standing wave field in one dimension (position ) is where is the acoustic pressure (the local deviation from the ambient pressure), and where is the speed of sound. Provided that the speed is a constant, not dependent on frequency (the dispersionless case), then the most general solution is where and are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one () traveling up the x-axis and the other () down the x-axis at the speed . The particular case of a sinusoidal wave traveling in one direction is obtained by choosing either or to be a sinusoid, and the other to be zero, giving where is the angular frequency of the wave and is its wave number. The derivation of the wave equation involves three steps: derivation of the equation of state, the linearized one-dimensional continuity equation, and the linearized one-dimensional force equation. The equation of state (ideal gas law) In an adiabatic process, pressure P as a function of density can be linearized to where C is some constant.