Editing SSC/SoundWaves (section)

====Wave Equation Derivation====
It is customary to combine these three relations to obtain a single, second-order partial-differential equation in terms of (any) one of the perturbation amplitudes.  We begin by using the third equation to replace <math>~P_1</math> in favor of <math>~\rho_1</math> in the second equation. This generates,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\rho_0 \frac{\partial \vec{v}}{\partial t}  + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, .
</math>
  </td>
</tr>
</table>
</div>
Taking the divergence of this equation gives,
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<tr>
  <td align="right">
<math>~
\rho_0 \frac{\partial}{\partial t}(\nabla\cdot \vec{v})  + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla^2 \rho_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, ;
</math>
  </td>
</tr>
</table>
</div>

while taking the time derivative of the first (''i.e.,'' the linearized continuity) equation gives,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial^2 \rho_1}{\partial t^2}  + \rho_0 \frac{\partial}{\partial t}(\nabla\cdot \vec{v})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
(Note that we have freely interchanged the order of the <math>~\nabla</math> and <math>~\partial/\partial t</math> operators because the spatial operator is not a function of time.  Also, as before, quantities having a subscript "0" have been pulled outside of both operators because, in this discussion, they have no time- or spatial-dependence.)  Finally, taking the difference between these last two relations produces the anticipated,

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<tr><td align="center">
<font color="#770000">'''Wave Equation'''</font>

<math>~
\frac{\partial^2 \rho_1}{\partial t^2}  - c_s^2 \nabla^2 \rho_1 = 0
</math>
</td></tr>
</table>
</div>
exhibiting the wave propagation speed,
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<math>~
c_s  = \sqrt{\biggl( \frac{dP}{d\rho} \biggr)_0} \, .
</math>
</div>
As derived, this wave equation describes, from an Eulerian (as opposed to Lagrangian) perspective, how the density perturbation, <math>~\rho_1(\vec{r},t)</math>, varies with time at any coordinate position.