Editing SSC/SoundWaves (section)

===Perturbation then Linearization of Equations===

Following [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>] &#8212; text in green is taken ''verbatum'' from their Chapter VIII (pp. 245-248) &#8212; we <FONT COLOR="#007700">begin by investigating small oscillations; an oscillatory motion of small amplitude in a compressible fluid is called a ''sound wave.''</FONT>  Given that <FONT COLOR="#007700">the relative changes in the fluid density and pressure are small</FONT>, in this ''Eulerian'' analysis where we are investigating how conditions vary with time at a fixed point in space, <math>~\vec{r}</math>, <FONT COLOR="#007700">we can write the variables {{Math/VAR_Pressure01}} and {{Math/VAR_Density01}} in the form</FONT>,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0 + \rho_1(\vec{r},t) \, ,</math>
  </td>
</tr>
</table>
</div>
<FONT COLOR="#007700">where <math>~\rho_0</math> and <math>~P_0</math> are the constant</FONT> (both in space and time) <FONT COLOR="#007700">equilibrium density and pressure, and <math>~\rho_1</math> and <math>~P_1</math> are their variations in the sound wave</FONT> <math>~(|\rho_1/\rho_0 | \ll 1, | P_1/P_0 | \ll 1)</math>.  <FONT COLOR="#007700">Since the oscillations are small</FONT> &#8212; and because we are assuming that the fluid is initially stationary <math>~(\mathrm{i.e.,}~\vec{v}_0 = 0)</math> &#8212; <FONT COLOR="#007700">the velocity</FONT> {{Math/VAR_VelocityVector01}} <FONT COLOR="#007700">is small also</FONT>.  In what follows, by definition, <math>~P_1</math>, <math>~\rho_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.

Substituting the expression for <math>~\rho</math> into the lefthand side of the continuity equation and <FONT COLOR="#007700">neglecting small quantities of the second order</FONT>, we have,
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<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} \, , 
</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 and, in the second line, <math>~\rho_0</math> has been pulled outside of the divergence operator because we have assumed that the initial equilibrium state is homogeneous.  Hence, we have (see, also, equation 63.2 of [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>]) the,
<div align="center">
<font color="#770000">'''Linearized Continuity Equation'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

Next, we note that <FONT COLOR="#007700">the term,</FONT>
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<table border="0" cellpadding="5">
<tr><td align="center">
<math>(\vec{v} \cdot \nabla)\vec{v} \, ,</math>
</td></tr>
</table>
</div>
<FONT COLOR="#007700">in Euler's equation may be neglected</FONT> because it is of second order in smallness.   Substituting the expressions for <math>~\rho</math> and <math>~P</math> into the righthand side of the Euler equation and <FONT COLOR="#007700">neglecting small quantities of the second order</FONT>, 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)</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[ \cancelto{0}{\nabla P_0} + \nabla P_1\biggr]
</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] \nabla P_1
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho_0} \nabla P_1 - \frac{1}{\rho_0^2} \cancelto{\mathrm{small}}{(\rho_1 \nabla P_1)} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in the first line, <math>~\nabla P_0</math> has been set to zero because we have assumed that the initial equilibrium state is homogeneous, and the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] has been used to obtain the expression on the righthand side of the second line.  Combining these simplification steps, we have (see, also, equation 63.3 of [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>]) 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 \, .
</math>
  </td>
</tr>
</table>
</div>
Ultimately, as emphasized in [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], <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>.

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, we have (see, also, equation 63.4 of [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>]) 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>