Editing SSC/PerspectiveReconciliation (section)

=Rosseland's Derivation of the Wave Equation=

Here we step through a derivation of the wave equation as has been presented by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf S. Rosseland (1969)]
in chapter 2 of his book titled, ''The Pulsation Theory of Variable Stars'' (see, especially his &sect;&sect; 2.1 &amp; 2.2, pp. 15-20), dropping all terms that account for nonadiabatic effects.

We begin with the set of [[PGE#Principal_Governing_Equations|principal governing equations]] that provides the foundation for practically all of our discussions in this H_Book, namely, the

<div align="center">
<span id="PGE:Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

{{Math/EQ_Continuity01}}



<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />

{{Math/EQ_Euler01}}


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

{{Math/EQ_FirstLaw02}} .


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Math/EQ_Poisson01}}
</div>


Because we are considering only spherically symmetric configurations, we will implement operators that reflect structural variations only in the radial coordinate.  Specifically, following a [[SSCpt1/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|parallel discussion of the principal governing equations that are appropriate for spherically symmetric configurations]], we have the,

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{\partial (r^2 v_r)}{\partial r}  \biggr] = 
\frac{d\rho}{dt} + \rho \biggl[ \frac{\partial v_r}{\partial r} + \frac{2v_r}{r} \biggr] = 
0 </math>



<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />

<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{d\Phi}{dr} </math><br />



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

{{Math/EQ_FirstLaw02}}


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math><br />
</div>

==Step 1:==
Following Rosseland, we adopt as a supplemental relation what we have [[SR/IdealGas#Consequential_Ideal_Gas_Relations|referred to elsewhere]] as,

<div align="center">
<span id="ConservingMass:Lagrangian"><font color="#770000">'''Form B'''</font></span><br />
of the Ideal Gas Equation of State,

{{Math/EQ_EOSideal02}}
</div>
where, for our purposes, we assume that {{Math/MP_AdiabaticIndex}} is independent of space and time.  Plugging the function, <math>~\epsilon(P,\rho)</math>, that is defined by this equation of state into the adiabatic form of the 1<sup>st</sup> law of thermodynamics leads to the relations (see, respectively, Rosseland's equations 2.8 and 2.9),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dP}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\gamma_g P}{\rho} \frac{d\rho}{dt} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P}{P_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_0}\biggr)^{\gamma_g} \, ,</math>
  </td>
</tr>
</table>
</div>

where <math>~P_0</math> and <math>~\rho_0</math> are independent of time.  We note for later use that, when the equation of continuity &#8212; as written in its original vector-operator form &#8212; is combined with this last differential form of the 1<sup>st</sup> law of thermodynamics, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dP}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \gamma_g P \nabla\cdot \vec{v} \, .</math>
  </td>
</tr>
</table>
</div>


==Step 2:==
As we have shown in a 
[[SSCpt2/SolutionStrategies#Technique_1|separate discussion of how to construct equilibrium, spherically symmetric configurations]], the Poisson equation can be integrated once to give an expression for the gravitational acceleration (see, also, Rosseland's equation 2.16) of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g \equiv \frac{d\Phi}{dr}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM_r}{r^2} \, .</math>
  </td>
</tr>
</table>
</div>

As is quite customary in this field of research, Rosseland substitutes <math>~g</math> for the radial gradient of the gravitational potential in the Euler equation &#8212; that is, he writes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dv_r}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho} \frac{\partial P}{\partial r} - g \, ,</math>
  </td>
</tr>
</table>
</div>

(see his equation 2.14 with nonadiabatic terms set to zero) &#8212; and, in so doing, considers that the Poisson equation has been solved.  This reduces the number of governing equations (as well as the number of unknown structural variables) to three.  Then Rosseland proceeds to point out that, when the radial pulsation of a spherically symmetric configuration is viewed from a Lagrangian perspective &#8212; that is, while riding along with a fluid element &#8212; the mass enclosed inside the radial position of each fluid element, <math>~M_r</math>, does not change with time.  Hence, the ''Lagrangian'' time-derivative of the gravitational acceleration is (see Rosseland's equation 2.17),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dg}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~GM_r \frac{d}{dt}\biggl(\frac{1}{r^2}\biggr) = - \frac{2GM_r}{r^3} \frac{dr}{dt} = - \frac{2gv_r}{r} \, .</math>
  </td>
</tr>
</table>
</div>

==Step 3:==
Next, Rosseland takes the ''Lagrangian'' time-derivative of the Euler equation, which gives (see Rosseland's equation 2.15 with nonadiabatic terms set to zero),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2v_r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{d}{dt}\biggl[\frac{1}{\rho} \frac{\partial P}{\partial r}\biggr] - \frac{dg}{dt} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\rho^2}\frac{d\rho}{dt} \biggl(\frac{\partial P}{\partial r}\biggr)
- \frac{1}{\rho} \frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] - \frac{dg}{dt} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho}\biggl(\frac{\partial P}{\partial r}\biggr)\biggl[ \frac{\partial v_r}{\partial r} + \frac{2v_r}{r}  \biggr]
- \frac{1}{\rho} \frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] + \frac{2gv_r}{r} \, .</math>
  </td>
</tr>
</table>
</div>
But, from the Euler equation, itself, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{1}{\rho} \biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{dv_r}{dt} + g \, .</math>
  </td>
</tr>
</table>
</div>
Hence,

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

<tr>
  <td align="right">
<math>~\frac{d^2v_r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{dv_r}{dt} + g \biggr] \biggl[ \frac{\partial v_r}{\partial r} + \frac{2v_r}{r}  \biggr]
- \frac{1}{\rho} \frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] + \frac{2gv_r}{r} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{dv_r}{dt} +g\biggr]\frac{\partial v_r}{\partial r}  
- \frac{1}{\rho} \frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] + \frac{4gv_r}{r}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

==Step 4:==
Finally, Rosseland sets out to swap the order of the time- and space-derivatives in the term,
<div align="center">
<math>~\frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] \, .</math>
</div>
But, whereas the ''partial'' time derivative commutes with the ''partial'' radial derivative, the total time derivative does not commute with the spatial derivative.  Generically, however, for any scalar variable, <math>~q</math>, we can swap between total and partial time derivatives via the operator relation,
<div align="center">
<math>~\frac{dq}{dt} = \frac{\partial q}{\partial t} + v_r \frac{\partial q}{\partial r} \, .</math>
</div>
Hence, if <math>~q</math> is replaced by <math>~P</math>, we can write,
<div align="center">
<math>~ v_r \biggl(\frac{\partial P}{\partial r}\biggr)  = \frac{dP}{dt}  - \frac{\partial P}{\partial t}\, ;</math>
</div>
while, if <math>~q</math> is replaced by <math>~\partial P/\partial r</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial t}\biggl( \frac{\partial P}{\partial r} \biggr) + v_r \frac{\partial }{\partial r} \biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) 
+ \frac{\partial }{\partial r} \biggl[ v_r \biggl( \frac{\partial P}{\partial r} \biggr)\biggr] 
- \biggl( \frac{\partial P}{\partial r} \biggr)\frac{\partial v_r}{\partial r} \, ,</math>
  </td>
</tr>
</table>
</div>
where, in making this last step, we have swapped the order of the ''partial'' derivatives in the first term on the right-hand side.  Combining these two relations and incorporating the form of Euler's equation that was highlighted in step #3 gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) 
+ \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  - \frac{\partial P}{\partial t} \biggr] 
+ \rho \biggl[  \frac{dv_r}{dt} + g \biggr]\frac{\partial v_r}{\partial r} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  \biggr] 
+ \rho \biggl[  \frac{dv_r}{dt} + g \biggr]\frac{\partial v_r}{\partial r} \, .</math>
  </td>
</tr>
</table>
</div>
Inserting this into the equation found at the end of step #3, then inserting the expression for <math>~dP/dt</math> derived in step #1 gives, 

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

<tr>
  <td align="right">
<math>~\frac{d^2v_r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  \biggr] + \frac{4gv_r}{r}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \gamma_g P \nabla\cdot \vec{v}\biggr] + \frac{4gv_r}{r}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
This matches Rosseland's equation (2.18) with the nonadiabatic terms set to zero.




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