Editing Cylindrical3D/Linearization (section)

===Linearizing Radial Component of Euler Equation===
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<math>~\frac{\partial {\dot\varpi}^'}{\partial t} + 
\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] 
-  \varpi ( { {\dot\varphi}_0 + {\dot\varphi}^'})^2 </math>
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<math>~=</math>
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<math>~- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial\varpi} - \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}</math>
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<math>~\Rightarrow~~~~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] 
-  \varpi ( {\dot\varphi}_0)^2 -  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')</math>
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<math>~=</math>
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<math>~
- \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi} 
- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0}  \biggr) 
- \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}
</math>
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<math>~\Rightarrow~~~~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
{\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} 
-  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
+ \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial\varpi}\biggr] 
+ \frac{\partial \Phi^'}{\partial \varpi}
</math>
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<math>~=</math>
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<math>~\biggl\{  \varpi ( {\dot\varphi}_0)^2 
- \frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi} 
- \frac{\partial \Phi_0}{\partial\varpi} \biggr\}
</math>
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<math>~\Rightarrow~~~~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
{\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} 
-  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
+ \biggl[ \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr] + \frac{\partial \Phi^'}{\partial \varpi}
</math>
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<math>~=</math>
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<math>~0 \, .
</math>
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This last expression has been obtained by recognizing that, in the next-to-last expression: (1) The terms inside the curly braces on the right-hand side collectively provide a statement of equilibrium (in the radial-coordinate direction) in the initial, unperturbed configuration and therefore the terms sum to zero; and (2) the terms inside square brackets on the left-hand side can be rewritten in a more compact form because we have adopted a polytropic equation of state to build the unperturbed initial equilibrium configuration and are examining only adiabatic perturbations with <math>~\gamma = (n+1)/n</math>, in which case,
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<math>~\frac{\nabla P_0}{P_0} = \frac{(n+1)}{n} \cdot \frac{\nabla \rho_0}{\rho_0} \, ,</math>
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&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
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<math>~\frac{P^'}{P_0} = \frac{\gamma \rho^'}{\rho_0} \, .</math>
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