Editing SSC/Perturbations (section)

====Entropy Conservation====
If, instead, we simply demand that the specific entropy remain constant in time, then the expression that relates <math>P</math> to <math>\rho</math> is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{P}{\rho^{\gamma_g}}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\exp\biggl[ \frac{s(\gamma_g - 1)}{\mathfrak{R}/\bar\mu}\biggr]\, .
</math>
  </td>
</tr>
</table>

Plugging the perturbed expressions for <math>P(m,t)</math> and <math>\rho(m,t)</math> into this expression gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\exp\biggl[ \frac{s(\gamma_g - 1)}{\mathfrak{R}/\bar\mu}\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \biggr\}
\biggl\{\rho_0(m) \biggl[1 + d(m) e^{i\omega t}  \biggr] \biggr\}^{-\gamma_g}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>P_0(m) \rho_0(m)^{-\gamma_g} 
\biggl[1 + p(m) e^{i\omega t}  \biggr]
\biggl[1 + d(m) e^{i\omega t}  \biggr]^{-\gamma_g} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[1 + p(m) e^{i\omega t}  \biggr]
\biggl[1 + d(m) e^{i\omega t}  \biggr]^{-\gamma_g} \, .
</math>
  </td>
</tr>
</table>
Now, according to the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]],

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

<tr>
  <td align="right">
<math>(1 \pm x)^n</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 ~\pm ~nx + \biggl[\frac{n(n-1)}{2!}\biggr]x^2
~\pm~ \biggl[\frac{n(n-1)(n-2)}{3!}\biggr]x^3
+ \biggl[\frac{n(n-1)(n-2)(n-3)}{4!}\biggr]x^4
~~\pm ~~ \cdots
</math>
&nbsp; &nbsp;&nbsp; for <math>~(x^2 < 1)</math>
  </td>
</tr>
</table>

Hence, we see that,

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

<tr>
  <td align="right">
<math>\biggl[ 1 + d(m) e^{i\omega t}\biggr]^{-\gamma_g}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
1 ~- ~\gamma_g d(m) e^{i\omega t} 
+ \biggl[\frac{\gamma_g(\gamma_g+1)}{2!}\biggr]\biggl[d(m) e^{i\omega t}\biggr]^2 \, .
</math>
  </td>
</tr>
</table>
Dropping terms of higher order than unity, the relationship between <math>p(m)</math> and <math>d(m)</math> becomes,

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

<tr>
  <td align="right">
<math>1</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math> 
\biggl[1 + p(m) e^{i\omega t}  \biggr]
\biggl[1 ~- ~\gamma_g d(m) e^{i\omega t} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
1 + p(m) e^{i\omega t} 
~- ~\gamma_g d(m) e^{i\omega t} 
~- ~\gamma_g \cancelto{\mathrm{small}}{d(m)p(m)} e^{2i\omega t}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ p(m)</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>  
\gamma_g d(m)  \, .
</math>
  </td>
</tr>
</table>

This is identical to the algebraic relationship between <math>p(m)</math> and <math>d(m)</math> that has been derived, [[#Adiabatic_form_of_the_First_Law_of_Thermodynamics|immediately above]].