Theory of sound waves
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab9f9d9f56b62c7ccf6c8e21d5f14e4cdad3db33)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/916d440edb0a1fb2c98abe6f7318c2b8315d9ee7)
Where
is the perturbed velocity of the fluid,
is the pressure of the fluid at rest,
is the perturbed pressure of the system as a function of space and time,
is the density of the fluid at rest, and
is the variance in the density of the fluid over space and time.
In the case that the velocity is irrotational (
), 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f04231df723be6dc89c09823e74d2ba2b64703)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4301271a376de1b33d0565de12f5bf16ac823341)