Editing SSC/StabilityEulerianPerspective (section)

====Euler Equation====
Next, we turn to the Euler equation and note that the term,
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<math>(\vec{v} \cdot \nabla)\vec{v} \, ,</math>
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may be altogether neglected because it is of second order in smallness.   Substituting the expressions for <math>~\rho</math>, <math>~P</math>, and <math>~\Phi</math> into the righthand side of the Euler equation and neglecting small quantities of the second order, we have,
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<math>~- \frac{1}{(\rho_0 + \rho_1)} \nabla (P_0 + P_1) - \nabla(\Phi_0 + \Phi_1)</math>
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<math>~=</math>
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<math>~-
\frac{1}{\rho_0} \biggl( 1 + \frac{\rho_1}{\rho_0} \biggr)^{-1} \biggl[ \nabla P_0 + \nabla P_1\biggr] - \nabla(\Phi_0 + \Phi_1)
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&nbsp;
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<math>~=</math>
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<math>~-
\frac{1}{\rho_0} \biggl[ 1 - \frac{\rho_1}{\rho_0} + \cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2} + \cdots \biggr] 
\biggl[ \nabla P_0 + \nabla P_1\biggr] - \nabla(\Phi_0 + \Phi_1)
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&nbsp;
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<math>~=</math>
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<math>~ - \cancelto{0}{\biggl[ \frac{1}{\rho_0} \nabla P_0 + \nabla \Phi_0 \biggr]}
-\frac{1}{\rho_0} \nabla P_1 - \nabla\Phi_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0
+ \frac{1}{\rho_0^2} \cancelto{\mathrm{small}}{(\rho_1 \nabla P_1)}  \, ,
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where, the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] has been used to obtain the expression on the righthand side of the second line and, in the last line, the sum of the first pair of terms has been set to zero because the initial configuration is assumed to be in equilibrium.  Combining these simplification steps, we have the,
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<font color="#770000">'''Linearized Euler Equation'''</font><br />
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<math>~
\frac{\partial \vec{v}}{\partial t}  
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<math>~=</math>
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<math>~
- \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0 - \nabla\Phi_1 \, .
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Ultimately, as emphasized in [[Appendix/References#LL75|LL75]], <FONT COLOR="#007700">the condition that the</FONT> linearized governing equations <FONT COLOR="#007700">should be applicable to the propagation of sound waves is that the velocity of the fluid particles in the wave should be small compared with the velocity of sound</FONT>, that is, <math>~|\vec{v}| \ll c_s</math>.
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