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 or particle velocityu as a function of position x and time. 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.
Provided that the speed is a constant, not dependent on frequency, 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 travelling up the x-axis and the other down the x-axis at the speed. The particular case of a sinusoidal wave travelling 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.
Derivation
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 In an adiabatic process, pressure P as a function of density can be linearized to where C is some constant. Breaking the pressure and density into their mean and total components and noting that : The adiabatic bulk modulus for a fluid is defined as which gives the result Condensation, s, is defined as the change in density for a given ambient fluid density. The linearized equation of state becomes The continuity equation in one dimension is Where u is the flow velocity of the fluid. Again the equation must be linearized and the variables split into mean and variable components. Rearranging and noting that ambient density changes with neither time nor position and that the condensation multiplied by the velocity is a very small number: Euler's Force equation is the last needed component. In one dimension the equation is: where represents the convective, substantial or material derivative, which is the derivative at a point moving with medium rather than at a fixed point. Linearizing the variables: Rearranging and neglecting small terms, the resultant equation becomes the linearized one-dimensional Euler Equation: Taking the time derivative of the continuity equation and the spatial derivative of the force equation results in: Multiplying the first by, subtracting the two, and substituting the linearized equation of state, The final result is where is the speed of propagation.
In three dimensions
Equation
Feynman provides a derivation of the wave equation for sound in three dimensions as where is the Laplace operator, is the acoustic pressure, and where is the speed of sound. 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 fieldvelocity potential which has the form and then derive the physical quantities particle velocity and acoustic pressure by the equations :
Solution
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.
Cartesian coordinates
Cylindrical coordinates
where the asymptotic approximations to the Hankel functions, when, are
Spherical coordinates
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.
