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. Propagating waves in a pre-defined direction can also be calculated using a 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.[1]
The wave equation describing a standing wave field in one dimension (position
x
{\partial2p\over\partialx2}-{1\overc2}{\partial2p\over\partialt2}=0,
where
p
c
Provided that the speed
c
p=f(ct-x)+g(ct+x)
where
f
g
f
g
c
f
g
p=p0\sin(\omegat\mpkx)
where
\omega
k
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)
PV=nRT
In an adiabatic process, pressure P as a function of density
\rho
P=C\rho
where C is some constant. Breaking the pressure and density into their mean and total components and noting that
C= | \partialP |
\partial\rho |
P-P0=\left(
\partialP | |
\partial\rho |
\right)(\rho-\rho0)
The adiabatic bulk modulus for a fluid is defined as
B=\rho0\left(
\partialP | |
\partial\rho |
\right)adiabatic
which gives the result
P-P0=B
\rho-\rho0 | |
\rho0 |
Condensation, s, is defined as the change in density for a given ambient fluid density.
s=
\rho-\rho0 | |
\rho0 |
The linearized equation of state becomes
p=Bs
P-P0
The continuity equation (conservation of mass) in one dimension is
\partial\rho | |
\partialt |
+
\partial | |
\partialx |
(\rhou)=0
Where u is the flow velocity of the fluid.Again the equation must be linearized and the variables split into mean and variable components.
\partial | |
\partialt |
(\rho0+\rho0s)+
\partial | |
\partialx |
(\rho0u+\rho0su)=0
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:
\partials | |
\partialt |
+
\partial | |
\partialx |
u=0
Euler's Force equation (conservation of momentum) is the last needed component. In one dimension the equation is:
\rho
Du | |
Dt |
+
\partialP | |
\partialx |
=0
where
D/Dt
Linearizing the variables:
(\rho0+\rho0s)\left(
\partial | |
\partialt |
+u
\partial | |
\partialx |
\right)u+
\partial | |
\partialx |
(P0+p)=0
Rearranging and neglecting small terms, the resultant equation becomes the linearized one-dimensional Euler Equation:
\rho | ||||
|
+
\partialp | |
\partialx |
=0
Taking the time derivative of the continuity equation and the spatial derivative of the force equation results in:
\partial2s | |
\partialt2 |
+
\partial2u | |
\partialx\partialt |
=0
\rho0
\partial2u | |
\partialx\partialt |
+
\partial2p | |
\partialx2 |
=0
Multiplying the first by
\rho0
-
\rho0 | |
B |
\partial2p | |
\partialt2 |
+
\partial2p | |
\partialx2 |
=0
The final result is
{\partial2p\over\partialx2}-{1\overc2}{\partial2p\over\partialt2}=0
where
c=\sqrt{
B | |
\rho0 |
Feynman[3] provides a derivation of the wave equation for sound in three dimensions as
\nabla2p-{1\overc2}{\partial2p\over\partialt2}=0,
where
\nabla2
p
c
A similar looking wave equation but for the vector field particle velocity is given by
\nabla2u -{1\overc2}{\partial2u \over\partialt2}=0
In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form
\nabla2\Phi-{1\overc2}{\partial2\Phi\over\partialt2}=0
u=\nabla\Phi
p=-\rho{\partial\over\partialt}\Phi
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
ei\omega
\omega=2\pif
p(r,t,k)=\operatorname{Real}\left[p(r,k)ei\omega\right]
k=\omega/c
p(r,k)=Ae\pm
(1) | |
p(r,k)=AH | |
0 |
(kr)+
(2) | |
BH | |
0 |
(kr)
where the asymptotic approximations to the Hankel functions, when
kr → infty
(1) | |
H | |
0 |
(kr)\simeq\sqrt{
2 | |
\pikr |
(2) | |
H | |
0 |
(kr)\simeq\sqrt{
2 | |
\pikr |
p(r,k)= | A |
r |
e\pm
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.