Editing SSC/StabilityEulerianPerspective (section)

===Perturbation then Linearization of Equations===

In this ''Eulerian'' analysis, we are investigating how conditions vary with time at a fixed point in space, <math>~\vec{r}</math>.  By analogy with [[SSC/SoundWaves#Perturbation_then_Linearization_of_Equations|our separate introductory  analysis of sound waves]], we will write the four  primary variables in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0(\vec{r}) + \rho_1(\vec{r},t) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cancelto{0}{\vec{v}_0} + \vec{v}_1(\vec{r},t) = \vec{v}(\vec{r},t) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0(\vec{r}) + P_1(\vec{r},t)  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Phi_0(\vec{r}) + \Phi_1(\vec{r},t)  \, ,</math>
  </td>
</tr>
</table>
</div>
where quantities with subscript "0" are initial values &#8212; independent of time, but not necessarily spatially uniform, and usually specified via some choice of an initial equilibrium configuration &#8212; and quantities with subscript "1" denote variations away from the initial state, which are assumed to be small in amplitude &#8212; for example, <math>~|\rho_1/\rho_0 | \ll 1</math> and <math>~| P_1/P_0 | \ll 1</math>.  As indicated, we will assume that the fluid configuration is initially stationary <math>~(\mathrm{i.e.,}~\vec{v}_0 = 0)</math> and, for simplicity, will not append the subscript "1" to the velocity perturbation.  It is to be understood, however, that the velocity, {{Math/VAR_VelocityVector01}}, is small also where, ultimately, this will mean <math>~|\vec{v}| \ll c_s</math>.  In what follows, by definition, <math>~P_1</math>, <math>~\rho_1</math>, <math>~\Phi_1</math>, and <math>~\vec{v}</math> are considered to be of first order in smallness, while products of these quantities are of second (or even higher) order in smallness.

====Continuity Equation====
Substituting the expression for <math>~\rho</math> into the lefthand side of the continuity equation and neglecting small quantities of the second order, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~~\frac{\partial}{\partial t} (\rho_0 + \rho_1) + \nabla\cdot [(\rho_0 + \rho_1)\vec{v}]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cancelto{0}{\frac{\partial \rho_0}{\partial t}} + \frac{\partial \rho_1}{\partial t}  
+ \nabla\cdot (\rho_0 \vec{v}) + \nabla\cdot\cancelto{\mathrm{small}}{(\rho_1\vec{v} )}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0 \, , 
</math>
  </td>
</tr>
</table>
</div>
where, in the first line, the first term on the righthand side has been set to zero because <math>~\rho_0</math> is independent of time.  Hence, we have the desired,
<div align="center">
<font color="#770000">'''Linearized Continuity Equation'''</font>
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
<!--
Note that, if we were to assume that the initial configuration is homogeneous, then we could set <math>~\nabla\rho_0 = 0</math> and drop the last term on the righthand side of this expression, retrieving the linearized continuity equation used in [[SSC/SoundWaves#Perturbation_then_Linearization_of_Equations|our introductory discussion of sound waves]].
-->

====Euler Equation====
Next, we turn to the Euler equation and note that the term,
<div align="center">
<table border="0" cellpadding="5">
<tr><td align="center">
<math>(\vec{v} \cdot \nabla)\vec{v} \, ,</math>
</td></tr>
</table>
</div>
may be altogether neglected because it is of second order in smallness.   Substituting the expressions for <math>~\rho</math>, <math>~P</math>, and <math>~\Phi</math> into the righthand side of the Euler equation and neglecting small quantities of the second order, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{1}{(\rho_0 + \rho_1)} \nabla (P_0 + P_1) - \nabla(\Phi_0 + \Phi_1)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\rho_0} \biggl( 1 + \frac{\rho_1}{\rho_0} \biggr)^{-1} \biggl[ \nabla P_0 + \nabla P_1\biggr] - \nabla(\Phi_0 + \Phi_1)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\rho_0} \biggl[ 1 - \frac{\rho_1}{\rho_0} + \cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2} + \cdots \biggr] 
\biggl[ \nabla P_0 + \nabla P_1\biggr] - \nabla(\Phi_0 + \Phi_1)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \cancelto{0}{\biggl[ \frac{1}{\rho_0} \nabla P_0 + \nabla \Phi_0 \biggr]}
-\frac{1}{\rho_0} \nabla P_1 - \nabla\Phi_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0
+ \frac{1}{\rho_0^2} \cancelto{\mathrm{small}}{(\rho_1 \nabla P_1)}  \, ,
</math>
  </td>
</tr>
</table>
</div>
where, the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] has been used to obtain the expression on the righthand side of the second line and, in the last line, the sum of the first pair of terms has been set to zero because the initial configuration is assumed to be in equilibrium.  Combining these simplification steps, we have the,
<div align="center">
<font color="#770000">'''Linearized Euler Equation'''</font><br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\frac{\partial \vec{v}}{\partial t}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0 - \nabla\Phi_1 \, .
</math>
  </td>
</tr>
</table>
</div>
<!--
Ultimately, as emphasized in [[Appendix/References#LL75|LL75]], <FONT COLOR="#007700">the condition that the</FONT> linearized governing equations <FONT COLOR="#007700">should be applicable to the propagation of sound waves is that the velocity of the fluid particles in the wave should be small compared with the velocity of sound</FONT>, that is, <math>~|\vec{v}| \ll c_s</math>.
-->

====First Law of Thermodynamics====
In a similar fashion, perturbing the variables in the barotropic equation of state leads to,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
P_0 + P_1  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K (\rho_0 + \rho_1)^{\gamma_\mathrm{g}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ P_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}} - K\rho_0^{\gamma_\mathrm{g}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K\rho_0^{\gamma_\mathrm{g}} \biggl[1 + 
\gamma_\mathrm{g}\biggl(\frac{\rho_1}{\rho_0} \biggr) + \frac{\gamma_\mathrm{g}(\gamma_\mathrm{g}-1)}{2} 
\cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2}
+ \cdots \biggr] - K\rho_0^{\gamma_\mathrm{g}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\gamma_\mathrm{g} \biggl( \frac{P_0}{\rho_0} \biggr) \rho_1 \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, as in [[SSC/SoundWaves#Perturbation_then_Linearization_of_Equations|our separate introductory discussion of sound waves]], we have the,
<div align="center">
<font color="#770000">'''Linearized Equation of State'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\, .
</math>
  </td>
</tr>
</table>
</div>

====Poisson Equation====
Finally, plugging the "perturbed" expressions for <math>~\Phi</math> and <math>~\rho</math> into the Poisson equation &#8212; which, by its very nature, is a linear equation &#8212; we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla^2 (\Phi_0 + \Phi_1)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G (\rho_0 + \rho_1)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \cancelto{0}{\nabla^2 \Phi_0 - 4\pi G \rho_0} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\nabla^2 \Phi_1 - 4\pi G \rho_1 \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in the second line, the sum of the pair of terms on the lefthand side has been set to zero because it is a self-contained representation of the Poisson equation for the initial unperturbed medium.  This gives us the desired,
<div align="center">
<font color="#770000">'''Linearized Poisson Equation'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla^2 \Phi_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G \rho_1 \, .
</math>
  </td>
</tr>
</table>
</div>