Editing Acoustic theory (section)

==Derivation for a medium at rest==
Starting with the Continuity Equation and the Euler Equation:
: <math>
  \begin{align}
     \frac{\partial \rho}{\partial t} +\nabla\cdot \rho\mathbf{v} & = 0 \\
     \rho\frac{\partial \mathbf{v}}{\partial t} + \rho(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p & = 0
  \end{align}
</math>

If we  take small perturbations of a constant pressure and density:
: <math>
  \begin{align}
     \rho & = \rho_0+\rho' \\
     p & = p_0 + p'
  \end{align}
</math>

Then the equations of the system are
: <math>
  \begin{align}
     \frac{\partial}{\partial t}(\rho_0+\rho') +\nabla\cdot (\rho_0+\rho')\mathbf{v} & = 0 \\
     (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla (p_0+p') & = 0
  \end{align}
</math>

Noting that the equilibrium pressures and densities are constant, this simplifies to
: <math>
  \begin{align}
     \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\nabla\cdot \rho'\mathbf{v} & = 0 \\
     (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0
  \end{align}
</math>

===A Moving Medium===
Starting with
: <math>
  \begin{align}
     \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{w}+\nabla\cdot \rho'\mathbf{w} & = 0 \\
     (\rho_0+\rho')\frac{\partial \mathbf{w}}{\partial t} + (\rho_0+\rho')(\mathbf{w}\cdot\nabla)\mathbf{w} + \nabla p' & = 0
  \end{align}
</math>

We can have these equations work for a moving medium by setting <math>\mathbf{w} = \mathbf{u} + \mathbf{v}</math>, where <math>\mathbf{u}</math> is the constant velocity that the whole fluid is moving at before being disturbed (equivalent to a moving observer) and <math>\mathbf{v}</math> is the fluid velocity.

In this case the equations look very similar:
: <math>
  \begin{align}
     \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' + \nabla\cdot \rho'\mathbf{v} & = 0 \\
     (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{u}\cdot\nabla)\mathbf{v} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0
  \end{align}
</math>

Note that setting <math>\mathbf{u} = 0</math> returns the equations at rest.