Acoustic wave equation

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.

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.

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.

[1] The wave equation describing a standing wave field in one dimension (position

the speed of sound, using subscript notation for the partial derivatives.

[2] Start with the ideal gas law where

Then, assuming the process is adiabatic, pressure

The conservation of mass and conservation of momentum can be written as a closed system of two equations[3]

This coupled system of two nonlinear conservation laws can be written in vector form as:

is a sufficiently small pertubation, i.e., any powers or products of

This results in the linearized equation

Likewise, small pertubations of the components of

is a constant, resulting in the alternative form of the linear acoustics equations:

After dropping the tilde for convenience, the linear first order system can be written as:

While, in general, a non-zero background velocity is possible (e.g. when studying the sound propagation in a constant-strenght wind), it will be assumed that

Then the linear system reduces to the second-order wave equation:

{\displaystyle p_{tt}=-K_{0}u_{xt}=-K_{0}u_{tx}=K_{0}\left({\frac {1}{\rho _{0}}}p_{x}\right)_{x}=c_{0}^{2}p_{xx},}

Hence, the acoustic equation can be derived from a system of first-order advection equations that follow directly from physics, i.e., the first integrals:

Conversely, given the second-order equation

is a constant, not dependent on frequency (the dispersionless case), then the most general solution is where

This may be pictured as the superposition of two waveforms of arbitrary profile, one (

The particular case of a sinusoidal wave traveling in one direction is obtained by choosing either

is the angular frequency of the wave and

Feynman[6] provides a derivation of the wave equation for sound in three dimensions as where

A similar looking wave equation but for the vector field particle velocity is given by In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity): 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 the asymptotic approximations to the Hankel functions, when

, are 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.