Editing SSC/StabilityEulerianPerspective (section)

===Governing Equations and Supplemental Relations===
We begin with the set of [[PGE#Principal_Governing_Equations|principal governing equations]] that provides the foundation for all of our discussions in this H_Book, namely, the

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<span id="ConservingMass:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Continuity Equation,

{{Math/EQ_Continuity02}}


<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,

{{Math/EQ_Euler02}}


<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{Math/EQ_FirstLaw02}} .


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Math/EQ_Poisson01}}
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As was done in our [[SSC/SoundWaves#Governing_Equations_and_Supplemental_Relations|separate, introductory discussion of sound waves]], we will assume that we are dealing with an ideal gas and supplement this set of equations with a barotropic (polytropic) equation of state,
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<math>~P = K\rho^{\gamma_\mathrm{g}}</math>&nbsp; &nbsp; &hellip; with &hellip; &nbsp; &nbsp; 
<math>\gamma_\mathrm{g} \equiv \frac{d\ln P_0}{d\ln \rho_0} = \frac{\rho_0}{P_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \, ,</math>
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which will ensure that the adiabatic form of the first law of thermodynamics is satisfied.  When we develop the linearized Euler equation, below, it will be useful to recognize that, assuming <math>~\gamma_\mathrm{g}</math> is uniform throughout the fluid, we can rewrite this last expression as,
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<math>~\frac{\nabla P_0}{P_0}</math>
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<math>~=</math>
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<math>~\biggl[\frac{\rho_0}{P_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] \frac{\nabla \rho_0}{\rho_0}</math>
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<math>~\Rightarrow ~~~ \nabla P_0</math>
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<math>~=</math>
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<math>~\biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_0 \, .</math>
  </td>
</tr>
</table>
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