Editing Acoustic theory (section)

{{Short description|Theory of sound waves}}
'''Acoustic theory''' is a scientific field that relates to the description of [[Sound#Waves|sound waves]]. It derives from [[fluid dynamics]]. See [[acoustics]] for the [[engineering]] approach.

For sound waves of any magnitude of a disturbance in velocity, pressure, and density we have
: <math>
  \begin{align}
     \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} + \nabla\cdot(\rho'\mathbf{v}) & = 0  \qquad \text{(Conservation of Mass)} \\
     (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0  \qquad \text{(Equation of Motion)}
  \end{align}
</math>

In the case that the fluctuations in velocity, density, and pressure are small, we can approximate these as
: <math>
  \begin{align}
     \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} & = 0 \\
     \frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p'& = 0
  \end{align}
</math>

Where <math>\mathbf{v}(\mathbf{x},t)</math> is the perturbed velocity of the fluid, <math>p_0</math> is the pressure of the fluid at rest, <math>p'(\mathbf{x},t)</math> is the perturbed pressure of the system as a function of space and time, <math>\rho_0</math> is the density of the fluid at rest, and <math>\rho'(\mathbf{x}, t)</math> is the variance in the density of the fluid over space and time.

In the case that the velocity is [[irrotational]] (<math>\nabla\times \mathbf{v} = 0</math>), we then have the acoustic wave equation that describes the system:
:<math>
   \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} - \nabla^2\phi = 0
 </math>

Where we have
:<math>
\begin{align}
   \mathbf{v} & = -\nabla \phi \\
c^2 & = (\frac{\partial p}{\partial \rho})_s\\
p' & = \rho_0\frac{\partial \phi}{\partial t}\\
\rho' & = \frac{\rho_0}{c^2}\frac{\partial \phi}{\partial t}
\end{align}
 </math>