Acoustic theory

Acoustic theory is a scientific field that relates to the description of 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

${\displaystyle {\begin{aligned}{\frac {\partial \rho '}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} +\nabla \cdot (\rho '\mathbf {v} )&=0\qquad {\text{(Conservation of Mass)}}\\(\rho _{0}+\rho '){\frac {\partial \mathbf {v} }{\partial t}}+(\rho _{0}+\rho ')(\mathbf {v} \cdot \nabla )\mathbf {v} +\nabla p'&=0\qquad {\text{(Equation of Motion)}}\end{aligned}}}$

In the case that the fluctuations in velocity, density, and pressure are small, we can approximate these as

${\displaystyle {\begin{aligned}{\frac {\partial \rho '}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} &=0\\{\frac {\partial \mathbf {v} }{\partial t}}+{\frac {1}{\rho _{0}}}\nabla p'&=0\end{aligned}}}$

Where ${\displaystyle \mathbf {v} (\mathbf {x} ,t)}$ is the perturbed velocity of the fluid, ${\displaystyle p_{0}}$ is the pressure of the fluid at rest, ${\displaystyle p'(\mathbf {x} ,t)}$ is the perturbed pressure of the system as a function of space and time, ${\displaystyle \rho _{0}}$ is the density of the fluid at rest, and ${\displaystyle \rho '(\mathbf {x} ,t)}$ is the variance in the density of the fluid over space and time.

In the case that the velocity is irrotational (${\displaystyle \nabla \times \mathbf {v} =0}$), we then have the acoustic wave equation that describes the system:

${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\partial t^{2}}}-\nabla ^{2}\phi =0}$

Where we have

${\displaystyle {\begin{aligned}\mathbf {v} &=-\nabla \phi \\c^{2}&=({\frac {\partial p}{\partial \rho }})_{s}\\p'&=\rho _{0}{\frac {\partial \phi }{\partial t}}\\\rho '&={\frac {\rho _{0}}{c^{2}}}{\frac {\partial \phi }{\partial t}}\end{aligned}}}$