Editing Apps/PapaloizouPringle84 (section)

===Seminal Work by Papaloizou &amp; Pringle===
Let's plug the three expressions for the components of the perturbed velocity into the linearized continuity equation and, as well, replace <math>~\rho^'</math> in favor of <math>~W^'</math> via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~W^' </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P^'}{\rho_0(\sigma + m{\dot\varphi}_0)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{ (\sigma + m{\dot\varphi}_0)} \biggl(\frac{\gamma P_0 \rho^' }{\rho_0^2}\biggr) \, .</math>
  </td>
</tr>

</table>
</div>

Also multiplying through by <math>~-i</math>, we obtain,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho^'[(\sigma + m{\dot\varphi}_0)]  + \frac{m\rho_0}{\varpi} (\varpi {\dot\varphi}^' )
- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi (i {\dot\varpi}^' ) \biggr] 
- \frac{\partial}{\partial z} \biggl[ \rho_0 (i{\dot{z}}^') \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~{\bar\sigma}^2 \biggl( \frac{\rho_0^2 W^'}{\gamma P_0  } \biggr)  
+ \frac{m\rho_0}{\varpi ({\bar\sigma}^2 - \kappa^2 )} \biggl\{ -  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr]  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi}{({\bar\sigma}^2 - \kappa^2 )} \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) mW^' \bar\sigma   \biggr] \biggr\} 
+ \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} \, .
</math>
  </td>
</tr>
</table>

</div>

Multiplying through by <math>~D^2 \equiv ({\bar\sigma}^2 - \kappa^2)^2</math> and reorganizing terms gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{D^2}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi}{D} \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2mW^' \bar\sigma }{2\varpi {\dot\varphi}_0}  \biggr)   \biggr] \biggr\} 
+ \frac{D m\rho_0}{\varpi} \biggl\{ -  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr]  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ D^2 \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
+ \biggl( \frac{D^2 {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } \biggr)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{D^2}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi}{D} \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2mW^' \bar\sigma }{2\varpi {\dot\varphi}_0}  \biggr)   \biggr] \biggr\} + \rho_0 D \biggl\{
- \frac{m^2W^' }{\varpi^2 }  \biggl[{\bar\sigma}^2 + \frac{\kappa^2 \varpi  }{ 2 {\dot\varphi}_0 } \biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr)   \biggr]  
- \frac{m\kappa^2 {\bar\sigma} }{ 2\varpi{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ D^2 \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
+ \biggl( \frac{D^2 {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } \biggr) \, .  
</math>
  </td>
</tr>
</table>

</div>

As can be confirmed by comparing it to equation (3.18) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image &#8212; this expression matches the 2<sup>nd</sup>-order, two-dimensional PDE that defines the eigenvalue problem discussed by Papaloizou and Pringle in their seminal 1984 paper.

<div align="center" id="EigenvaluePP84">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equation (3.18) extracted without modification from p. 726 of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou &amp; Pringle (1984)]<p></p>
"''The dynamical stability of differentially rotating discs with constant specific angular momentum''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 208, pp. 721-750 &copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="center">
[[File:PP84Eq3.18.png|500px|center|Papaloizou and Pringle (1984, MNRAS, 208, 721)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>

After dividing through by <math>~D^2</math> and, again, rearranging terms, we also have,


<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{ {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ 
\frac{\rho_0 \varpi  {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr\}
-\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{\frac{\rho_0 \varpi}{D} \biggl( \frac{\kappa^2mW^' \bar\sigma }{2\varpi {\dot\varphi}_0}  \biggr)   \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{\rho_0}{D} \biggl\{
\frac{m^2 {\bar\sigma}^2 W^' }{\varpi^2 } \biggr\}
+ \frac{\rho_0}{D} \biggl\{
\frac{m^2W^' }{\varpi^2 }  \biggl[\frac{\kappa^2 \varpi  }{ 2 {\dot\varphi}_0 } \biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr)   \biggr]  \biggr\}
+ \frac{\rho_0}{D} \biggl\{\frac{m\kappa^2 {\bar\sigma} }{ 2\varpi{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ 
\frac{\rho_0 \varpi  {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr\}
+ \biggl\{ \frac{\rho_0 m^2 {\bar\sigma}^2 W^' }{\varpi^2 D} \biggr\}
- \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0 \kappa^2m }{2 {\dot\varphi}_0 D} \biggl(  W^' \bar\sigma \biggr)   \biggr] 
+ \frac{\rho_0 \kappa^2 m}{2 {\dot\varphi}_0 D} \biggl[ \frac{W^' }{\varpi} \cdot \frac{\partial (m{\dot\varphi}_0)}{\partial\varpi}   \biggr]
+ \frac{\kappa^2 m \rho_0}{2{\dot\varphi}_0 D} \biggl[\frac{ {\bar\sigma} }{\varpi} \cdot \frac{\partial W^'}{\partial \varpi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ 
\frac{\rho_0 \varpi  {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr\}
+ \biggl\{ \frac{\rho_0 m^2 {\bar\sigma}^2 W^' }{\varpi^2 D} \biggr\}
- \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0 \kappa^2m }{2 {\dot\varphi}_0 D} (  W^' \bar\sigma ) \biggr] 
+ \frac{\rho_0 \kappa^2 m}{2 {\dot\varphi}_0 D} \biggl[ \frac{1}{\varpi} \cdot \frac{\partial (W^' \bar\sigma)}{\partial\varpi}   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl( 
\frac{\rho_0 \varpi  {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr)
+ \frac{\rho_0 m^2 {\bar\sigma}^2 W^' }{\varpi^2 D} 
- \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial W^'}{\partial z} \biggr) 
- \frac{m W^' \bar\sigma}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{\varpi D} \biggl( \frac{\varpi \kappa^2 }{2 {\dot\varphi}_0 } \biggr)\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Here, it is advantageous to note that, in place of the [[#epicyclic|definition of the (square of the) epicyclic frequency provided above]], we could have equally well written,
<div align="center" id="epicyclic2">
<math>~\kappa^2 = \frac{1}{\varpi^3} \frac{d j_0^2}{d \varpi} \, ,</math>
</div>
where, <math>~j_0(\varpi)\equiv \varpi^2{\dot\varphi}_0(\varpi)</math> is a function that specifies how the fluid's specific angular momentum varies with radius in the initial, unperturbed, equilibrium configuration.  (See our related discussion of [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''Simple Rotation Profiles'']].)  From this relation, we recognize as well that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dj_0}{d\varpi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\kappa^2 \varpi^3}{2j_0} = \frac{\kappa^2 \varpi}{2{\dot\varphi}_0} \, .</math>
  </td>
</tr>
</table>
</div>
So the last term inside the square brackets of our expression could meaningfully be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\rho_0}{\varpi D} \biggl( \frac{\varpi \kappa^2 }{2 {\dot\varphi}_0 } \biggr)\biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~\rightarrow~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\biggl[ \frac{\rho_0}{\varpi D} \biggl( \frac{dj_0}{d\varpi} \biggr)\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

If we let <math>~h^'</math> represent the radial derivative of the specific angular momentum in the initial, unperturbed, equilibrium configuration (Papaloizou &amp; Pringle use "h" instead of "j" to denote the specific angular momentum, and they use a prime to denote differentiation with respect to the radial coordinate), we see that our derived expression matches equation (2.19) of [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985)] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image.  Clearly this mathematical definition of the eigenvalue problem discussed by PP85 is fundamentally the same as the one introduced and discussed in PP84.  

<div align="center" id="EigenvaluePP85">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equation (2.19) extracted without modification from p. 803 of [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985)]<p></p>
"''The dynamical stability of differentially rotating discs. II''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 213, pp. 799-820 &copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="center">
[[File:PP85Eq2.19.png|500px|center|Papaloizou and Pringle (1985, MNRAS, 213, 799)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>