Editing SSC/Stability/Isothermal (section)

====Yabushita (1968)====
The linearized wave equation that {{ Yabushita68full }} used to examine the radial pulsation modes of pressure-truncated isothermal spheres is displayed in the following, boxed-in image: 

<div align="center" id="Yabushita68">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equation extracted from p. 111 of <br />{{ Yabushita68figure }}
</td></tr>
<tr>
  <td align="left">
<!-- [[File:Yabushita68WaveEq.png|500px|center|Yabushita (1968)]] -->

<table border="0" align="center" cellpadding="5" width="100%">
<tr>
  <td align="right" width="65%">
<math>
\frac{\partial^2}{\partial t^2} \delta\rho
- \nabla^2\delta p - 8\pi G \bar\rho \delta\rho
-\nabla\bar\rho \cdot \nabla\delta\Phi - \nabla\delta\rho\cdot \nabla\Phi
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left">
<math>
0 \, .
</math>
  </td>
  <td align="right" width="5%">(2.12)</td>
</tr>
</table>

  </td>
</tr>
</table>
</div>

This equation can be obtained straightforwardly through a strategic combination of three of the following four linearized principal governing equations that we have derived in our [[SSC/StabilityEulerianPerspective#Summary_and_Combinations|accompanying, broad introductory discussion]] of linear stability analyses, namely,

<div align="center">
<table border="1" cellpadding="15">
<tr><td align="center">
<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0 = 0 ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />
<math>
\frac{\partial \vec{v}}{\partial t}  = - \nabla\Phi_1 - \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0  \, ,
</math><br />

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

<math>
P_1 = \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\, ,
</math>

<font color="#770000">'''Linearized'''</font><br />
<font color="#770000">'''Poisson Equation'''</font><br />

<math>
\nabla^2 \Phi_1 = 4\pi G \rho_1\, .
</math>
</td></tr>
</table>
</div>

Taking the partial time-derivative of the linearized equation of continuity gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>- \nabla\cdot \frac{\partial \vec{v}}{\partial t} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\rho_0}\frac{\partial^2 \rho_1}{\partial t^2}  + \frac{\nabla\rho_0}{\rho_0} \cdot \frac{\partial\vec{v}}{\partial t}  \, ;</math>
  </td>
</tr>
</table>
</div>
and, taking the divergence of the linearized Euler equation gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-\nabla\cdot \frac{\partial \vec{v}}{\partial t} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\nabla^2 \Phi_1 + \nabla\cdot \biggl[\frac{1}{\rho_0} \nabla P_1\biggr]  - \nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Combining the two, then making two substitutions using (1) the linearized Poisson equation and (2) the linearized Euler equation, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\partial^2 \rho_1}{\partial t^2}  + \nabla\rho_0 \cdot \frac{\partial\vec{v}}{\partial t} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \nabla^2 \Phi_1 + \rho_0 \nabla\cdot \biggl[\frac{1}{\rho_0} \nabla P_1\biggr]  - \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
\frac{\partial^2 \rho_1}{\partial t^2}  + \nabla\rho_0 \cdot \biggl[ - \nabla\Phi_1 - \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi G \rho_0 \rho_1 + \nabla^2 P_1  + \rho_0 \nabla P_1 \cdot \nabla \biggl(\frac{1}{\rho_0} \biggr)  
- \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Rearranging terms, and using the replacement ''equilibrium'' relation, <math>\nabla P_0 = - \rho_0\nabla\Phi_0</math>, gives,

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

<tr>
  <td align="right">
<math>
\frac{\partial^2 \rho_1}{\partial t^2}  - \nabla^2 P_1 - 4\pi G \rho_0 \rho_1  - \nabla\rho_0\cdot\nabla\Phi_1 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\nabla\rho_0}{\rho_0} \cdot \biggl[  \nabla P_1 + \rho_1 \nabla \Phi_0 \biggr] 
+ \rho_0 \nabla P_1 \cdot \nabla \biggl(\frac{1}{\rho_0} \biggr)  
+ \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0} \nabla \Phi_0 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\nabla\rho_0}{\rho_0} \cdot \biggl[  \nabla P_1 \biggr] + \frac{\rho_1}{\rho_0} \biggl[ \nabla\rho_0\cdot \nabla \Phi_0 \biggr]
- \frac{1}{\rho_0} \nabla P_1 \cdot \nabla \rho_0  
+ \rho_0 \nabla \Phi_0 \cdot \nabla \biggl[ \frac{\rho_1}{\rho_0} \biggr] 
+ \rho_1\nabla^2  \Phi_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\rho_1}{\rho_0} \biggl[ \nabla\rho_0\cdot \nabla \Phi_0 \biggr] - \frac{\rho_1}{\rho_0} \biggl[ \nabla \Phi_0 \cdot \nabla\rho_0\biggr]
+ \nabla \Phi_0 \cdot \nabla  \rho_1
+ 4\pi G \rho_0 \rho_1 
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{\partial^2 \rho_1}{\partial t^2}  - \nabla^2 P_1 - 8\pi G \rho_0 \rho_1  - \nabla\rho_0\cdot\nabla\Phi_1 
- \nabla \Phi_0 \cdot \nabla  \rho_1
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>
</div>
This is identical to equation (2.12) of {{ Yabushita68 }}.  Letting <math>t \rightarrow (4\pi G \rho_c)^{-1 / 2} \tau</math> and noting that <math>\rho_c = P_c/c_s^2</math>, we also have,

<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>
4\pi G \rho_c^2 ~\biggl\{
\frac{\partial^2}{\partial \tau^2}\biggl(\frac{\rho_1}{\rho_c}\biggr)  - \frac{1}{4\pi G \rho_c^2}\nabla^2 P_1 - 2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{\rho_1}{\rho_c}\biggr)
- \frac{1}{4\pi G \rho_c^2} \nabla\rho_0\cdot\nabla\Phi_1 
- \frac{1}{4\pi G \rho_c^2} \nabla \Phi_0 \cdot \nabla  \rho_1
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial^2}{\partial \tau^2}\biggl(\frac{\rho_1}{\rho_c}\biggr)  - \frac{c_s^2}{4\pi G \rho_c}\nabla^2 \biggl(\frac{P_1}{P_c}\biggr) - 2\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{\rho_1}{\rho_c}\biggr)  
- \frac{c_s^2}{4\pi G \rho_c} \nabla\biggl( \frac{\rho_0}{\rho_c}\biggr) \cdot \nabla \biggl( \frac{\Phi_1 }{c_s^2}\biggr)
- \frac{c_s^2}{4\pi G \rho_c} \nabla \biggl( \frac{\Phi_0}{c_s^2}\biggr) \cdot \nabla \biggl(\frac{ \rho_1}{\rho_c} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial^2}{\partial \tau^2}\biggl(\frac{\rho_1}{\rho_c}\biggr)  - \nabla_\xi^2 \biggl(\frac{P_1}{P_c}\biggr) - 2\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{\rho_1}{\rho_c}\biggr)  
- \nabla_\xi\biggl( \frac{\rho_0}{\rho_c}\biggr) \cdot \nabla_\xi \biggl( \frac{\Phi_1 }{c_s^2}\biggr)
- \nabla_\xi \biggl( \frac{\Phi_0}{c_s^2}\biggr) \cdot \nabla_\xi \biggl(\frac{ \rho_1}{\rho_c} \biggr) \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in this last step, we have switched from the radial coordinate, <math>r</math>, to the dimensionless coordinate, <math>\xi \equiv r (4\pi G\rho_c/c_s^2)^{1 / 2}</math>.  This matches equation (2.15) of {{ Yabushita68 }}.  

Now, in principle, we can rewrite this linearized wave equation entirely in terms of the density perturbation by recognizing that, from the isothermal equation of state,
<div align="center">
<math>\frac{P_1}{P_c} = \frac{\rho_1}{\rho_c}</math>;
</div>
and from the Poisson equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla^2\Phi_1 = 4\pi G \rho_1</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>\nabla_\xi^2 \biggl( \frac{\Phi_1}{c_s^2} \biggr) = \frac{\rho_1}{\rho_c} \, .</math>
  </td>
</tr>
</table>
</div>
<span id="gDefinition">But this does not work directly because our</span> just-derived, governing linearized wave equation contains the term, <math>\nabla_\xi \Phi_1</math>, rather than, <math>\nabla_\xi^2 \Phi_1</math>.  Instead, following the lead of [http://adsabs.harvard.edu/abs/1957ZA.....42..263E Ebert (1957)], {{ Yabushita68 }} rewrites all of the perturbed variables in terms of a new function, <math>g(\xi)</math>, defined such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>g e^{i\omega t}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\xi^2 \frac{d}{d\xi}\biggl( \frac{\Phi_1}{c_s^2}\biggr)</math>
  </td>
</tr>
</table>
</div>
in which case, from the Poisson equation, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\rho_1}{\rho_c} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\xi^2} \frac{d}{d\xi} \biggl[\xi^2 \frac{d}{d\xi}\biggl( \frac{\Phi_1}{c_s^2} \biggr)\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\xi^2} \frac{dg}{d\xi} ~e^{i\omega t} \, .</math>
  </td>
</tr>
</table>
</div>
This matches both equation (12) of [http://adsabs.harvard.edu/abs/1957ZA.....42..263E Ebert (1957)], and equation (2.17) of {{ Yabushita68 }}.  The governing wave equation therefore becomes,

<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>
-\omega^2 \biggl[\frac{1}{\xi^2} \frac{dg}{d\xi} \biggr]  - \frac{1}{\xi^2}\frac{d}{d\xi} \biggl[ \xi^2 \frac{d}{d\xi}\biggl( \frac{1}{\xi^2} \frac{dg}{d\xi} \biggr) \biggr]
- 2\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl[ \frac{1}{\xi^2} \frac{dg}{d\xi} \biggr]   
- \nabla_\xi\biggl( \frac{\rho_0}{\rho_c}\biggr) \cdot \frac{g}{\xi^2}
- \nabla_\xi \biggl( \frac{\Phi_0}{c_s^2}\biggr) \cdot \frac{d}{d\xi} \biggl[ \frac{1}{\xi^2} \frac{dg}{d\xi}\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Finally, inserting the background, equilibrium structural profiles, in particular,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\rho_0}{\rho_c} = e^{-\psi}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>\frac{\Phi_0}{c_s^2} = \psi \, ,</math>
  </td>
</tr>
</table>
</div>
[<font color="red">NOTE: </font> Here, and throughout this H_Book, our adopted sign convention for <math>\psi</math> is opposite that adopted by {{ Yabushita68 }}; see is equations (2.5) and (2.6)] we have,

<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>
-\omega^2 \biggl[\frac{1}{\xi^2} \frac{dg}{d\xi} \biggr]  
- \frac{1}{\xi^2}\frac{d}{d\xi} \biggl[ \xi^2 \frac{d}{d\xi}\biggl( \frac{1}{\xi^2} \frac{dg}{d\xi} \biggr) \biggr]
+ e^{-\psi } \biggl[ - \frac{2}{\xi^2} \frac{dg}{d\xi}    + \frac{g}{\xi^2}\frac{d\psi}{d\xi} \biggr]
- \frac{d\psi}{d\xi} \cdot \frac{d}{d\xi} \biggl[ \frac{1}{\xi^2} \frac{dg}{d\xi}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\xi^2} \biggl\{
-\omega^2 \biggl[\frac{dg}{d\xi} \biggr]  
- \frac{d}{d\xi} \biggl[ \xi^2 \frac{d}{d\xi}\biggl( \frac{1}{\xi^2} \frac{dg}{d\xi} \biggr) \biggr]
+ e^{-\psi} \biggl[ -2\frac{dg}{d\xi}    + g \frac{d\psi}{d\xi} \biggr]
- \xi^2 \frac{d\psi}{d\xi} \cdot \frac{d}{d\xi} \biggl[ \frac{1}{\xi^2} \frac{dg}{d\xi}\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\xi^2} \biggl\{
-\omega^2 \biggl[\frac{dg}{d\xi} \biggr]  + \frac{2}{\xi} \frac{d^2g}{d\xi^2} - \frac{2}{\xi^2} \frac{dg}{d\xi} - \frac{d^3g}{d\xi^3} 
+ e^{-\psi} \biggl[ -2\frac{dg}{d\xi}    + g \frac{d\psi}{d\xi} \biggr]
- \frac{d\psi}{d\xi} \biggl[ - \frac{2}{\xi} \frac{dg}{d\xi} + \frac{d^2g}{d\xi^2}\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\omega^2 ~\frac{dg}{d\xi}  + \frac{d^3g}{d\xi^3}  
+  \frac{d^2g}{d\xi^2}\biggl[- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggl] 
+ \frac{dg}{d\xi} \biggl[ \frac{2}{\xi^2} - \frac{2}{\xi} \cdot \frac{d\psi}{d\xi} + 2e^{-\psi} \biggr]
- g \biggl[ e^{-\psi} \frac{d\psi}{d\xi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d^3g}{d\xi^3} - B_1(\xi) \frac{d^2 g}{d\xi^2} + [B_2(\xi) + \omega^2] \frac{dg}{d\xi} + B_3(\xi) g \, ,</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>B_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{2}{\xi} - \frac{d\psi}{d\xi} \biggl] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\biggl[ \frac{1}{\xi^2} - \frac{1}{\xi} \cdot \frac{d\psi}{d\xi} + e^{-\psi} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \biggl[ e^{-\psi} \frac{d\psi}{d\xi} \biggr]</math>
  </td>
</tr>
</table>
</div>
Taking into account that our sign convention on <math>~\psi</math> is opposite to that adopted by  [http://adsabs.harvard.edu/abs/1957ZA.....42..263E Ebert (1957)], this last form of the governing wave equation matches his eqs. (13) and (14) when his parameter, <math>\alpha</math>, is set to unity (isothermal condition) and the variable substitution, <math>\lambda \leftrightarrow i\omega</math>, is made.

Now, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d}{d\xi} \biggl[ \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) \biggl]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2g}{d\xi^2}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggl)
+
\biggl(\frac{2}{\xi^2} +\frac{d^2\psi}{d\xi^2} \biggr) \frac{dg}{d\xi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
\frac{d^2g}{d\xi^2}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggl)
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{d\xi} \biggl[ \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) \biggl]
-
\frac{dg}{d\xi}\biggl(\frac{2}{\xi^2} +\frac{d^2\psi}{d\xi^2} \biggr)  \, ,
</math>
  </td>
</tr>
</table>
</div>
we can rewrite the governing wave equation as,

<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}{d\xi} \biggl\{
\omega^2 g + \frac{d^2g}{d\xi^2} 
+  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) 
\biggr\}
+ \frac{dg}{d\xi} \biggl[ \frac{2}{\xi^2} - \frac{2}{\xi} \cdot \frac{d\psi}{d\xi} + 2e^{-\psi} \biggr]
- \frac{dg}{d\xi}\biggl(\frac{2}{\xi^2} +\frac{d^2\psi}{d\xi^2} \biggr)  
+ g \frac{d}{d\xi} \biggl(e^{-\psi} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{d}{d\xi} \biggl\{
\frac{d^2g}{d\xi^2} 
+  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) +
\omega^2 g \biggr\}
+ \frac{dg}{d\xi} \biggl[ - \frac{2}{\xi} \cdot \frac{d\psi}{d\xi} + 2e^{-\psi} - \frac{d^2\psi}{d\xi^2} \biggr]
+ \frac{d}{d\xi} \biggl(ge^{-\psi} \biggr) - e^{-\psi} \frac{dg}{d\xi} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{d}{d\xi} \biggl\{
\frac{d^2g}{d\xi^2} 
+  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) +
g(e^{-\psi} + \omega^2) \biggr\}
+ \frac{dg}{d\xi} \biggl[ - \frac{2}{\xi} \cdot \frac{d\psi}{d\xi} + e^{-\psi} - \frac{d^2\psi}{d\xi^2} \biggr]
\, .
</math>
  </td>
</tr>
</table>
</div>

Finally, we recognize that the last term in this expression drops out because, according to the isothermal Lane-Emden equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-\frac{d^2\psi}{d\xi^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2}{\xi} \cdot \frac{d\psi}{d\xi} - e^{-\psi}  \, .</math>
  </td>
</tr>
</table>
</div>
So, the governing wave equation becomes,

<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}{d\xi} \biggl\{
\frac{d^2g}{d\xi^2} 
+  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) +
g(e^{-\psi} + \omega^2) \biggr\}
\, ,
</math>
  </td>
</tr>
</table>
</div>
which can be integrated once to give, what we will refer to as the,
<div align="center" id="Yabushita68LAWE">
<font color="maroon"><b>Yabushita68 Isothermal LAWE</b></font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> 
\frac{d^2g}{d\xi^2} +  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) + g e^{-\psi}  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
C_0 - g \omega^2
\, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>C_0</math> is the integration constant.  Once again, taking into account the different adopted sign on <math>\psi</math>, acknowledging the variable substitution, <math>\lambda \leftrightarrow i\omega</math>, and considering only an isothermal equation of state <math>(\gamma = 1)</math>, we recognize that this is precisely the same form of the governing wave equation that appears as equation (2.19) of {{ Yabushita68 }}.