Wikipedia / Acoustic wave equation
Wikipedia / 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. a standing wavefield. The equation describes the evolution of acoustic pressure or particle velocity as a function of position 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.
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.
Definition in one dimension
The wave equation) is
is the acoustic pressure (the local deviation from the ambient pressure the speed of sound, using subscript notation for the partial derivatives.
Derivation
Start with the ideal gas law
the absolute temperature of the gas and specific gas constant. Then, assuming the process is adiabatic.
The conservation of mass and conservation of momentum can be written as a closed system of two equations
To linearize this equation, let can be discarded. Hence, the taylor expansion is the bulk modulus the speed of sound.
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 are similar.
Solution
is a constant, not dependent on frequency (the dispersionless case), then the most general solution is
are any two twice-differentiable functions. This may be pictured as the superposition to be a sinusoid, and the other to be zero, giving
.
is the angular frequency is its wave number.
In three dimensions
Equation
Feynman provides a derivation of the wave equation for sound in three dimensions as
is the Laplace operator is the acoustic pressure 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 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):
,
.
Solution
The following solutions are obtained by separation of variables in different coordinate systems. They are phasor is the angular frequency. The explicit time dependence is given by
is the wave number.
Cartesian coordinates
.
Cylindrical coordinates
.
where the asymptotic approximations to the Hankel functions, 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.