Editing SSC/StabilityEulerianPerspective (section)

====Continuity Equation====
Substituting the expression for <math>~\rho</math> into the lefthand side of the continuity equation and neglecting small quantities of the second order, we have,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~~\frac{\partial}{\partial t} (\rho_0 + \rho_1) + \nabla\cdot [(\rho_0 + \rho_1)\vec{v}]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cancelto{0}{\frac{\partial \rho_0}{\partial t}} + \frac{\partial \rho_1}{\partial t}  
+ \nabla\cdot (\rho_0 \vec{v}) + \nabla\cdot\cancelto{\mathrm{small}}{(\rho_1\vec{v} )}
</math>
  </td>
</tr>

<tr>
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&nbsp;
  </td>
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<math>~\approx</math>
  </td>
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<math>~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0 \, , 
</math>
  </td>
</tr>
</table>
</div>
where, in the first line, the first term on the righthand side has been set to zero because <math>~\rho_0</math> is independent of time.  Hence, we have the desired,
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<font color="#770000">'''Linearized Continuity Equation'''</font>
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<tr>
  <td align="right">
<math>~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
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Note that, if we were to assume that the initial configuration is homogeneous, then we could set <math>~\nabla\rho_0 = 0</math> and drop the last term on the righthand side of this expression, retrieving the linearized continuity equation used in [[SSC/SoundWaves#Perturbation_then_Linearization_of_Equations|our introductory discussion of sound waves]].
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