Editing SSC/SoundWaves (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, except, because we are ignoring the effects of self gravity, <math>~\nabla\Phi</math> is set to zero in the Euler equation and we drop the Poisson equation altogether.  Specifically, the relevant set of governing equations is, 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>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P </math>


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

{{Math/EQ_FirstLaw02}} .
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We supplement this set of equations with an ideal gas equation of state, specifically,
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{{Math/EQ_EOSideal02}} ,
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in which case the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics may be written as,
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<math>
\rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 \, .
</math>
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This, in turn implies,
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<math>
\frac{d\ln P}{d\ln\rho} = \gamma_\mathrm{g} \, ,
</math>
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which we will enforce by adopting the 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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