Editing SSC/SoundWaves (section)

=Sound Waves=
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<font size="-1">[[H_BookTiledMenu#Stability_Analysis|<b>Relationship<br />to<br />Sound Waves</b>]]</font>
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A standard technique that is used throughout astrophysics to test the stability of self-gravitating fluids involves ''perturbing'' physical variables away from their initial (usually equilibrium) values then ''linearizing'' each of the principal governing equations before seeking solutions describing the time-dependent behavior of the variables that simultaneously satisfy all of the equations.  When the effects of the fluid's self gravity are ignored and this analysis technique is applied to an initially homogeneous medium, the combined set of ''linearized'' governing equations generates a [http://en.wikipedia.org/wiki/Wave_equation wave equation] &#8212; whose ''general'' properties are well documented throughout the mathematics and physical sciences literature &#8212; that, specifically in our case, governs the propagation of sound waves.  It is quite advantageous, therefore, to examine how the wave equation is derived in the context of an analysis of sound waves before applying the standard perturbation &amp; linearization technique to inhomogeneous and self-gravitating fluids.

In what follows, we borrow heavily from Chapter VIII of [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], as it provides an excellent introductory discussion of sound waves.

==Assembling the Key Relations==

===Governing Equations and Supplemental Relations===
We begin with the set of [[PGE#Principal_Governing_Equations|principal governing equations]] that provides the foundation for all of our discussions in this H_Book, except, because we are ignoring the effects of self gravity, <math>~\nabla\Phi</math> is set to zero in the Euler equation and we drop the Poisson equation altogether.  Specifically, the relevant set of governing equations is, the

<div align="center">
<span id="ConservingMass:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Continuity Equation,

{{Math/EQ_Continuity02}}


<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,

<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P </math>


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

{{Math/EQ_FirstLaw02}} .
</div>

We supplement this set of equations with an ideal gas equation of state, specifically,
<div align="center">
{{Math/EQ_EOSideal02}} ,
</div>
in which case the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics may be written as,
<div align="center">
<math>
\rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 \, .
</math>
</div>

This, in turn implies,
<div align="center">
<math>
\frac{d\ln P}{d\ln\rho} = \gamma_\mathrm{g} \, ,
</math>
</div>
which we will enforce by adopting the barotropic (polytropic) equation of state,
<div align="center">

<math>~P = K\rho^{\gamma_\mathrm{g}}</math>&nbsp; &nbsp; &hellip; with &hellip; &nbsp; &nbsp; 
<math>\gamma_\mathrm{g} \equiv \frac{d\ln P_0}{d\ln \rho_0} = \frac{\rho_0}{P_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \, .</math>
</div>

===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,
<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} \, , 
</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>
<div align="center">
<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>

===Summary===
In summary, the following three linearized equations govern the time-dependent physical relationship between the three perturbation amplitudes <math>~P_1(\vec{r},t)</math>, <math>~\rho_1(\vec{r},t)</math> and <math>~\vec{v}(\vec{r},t)</math> in the context of sound waves:

<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} = 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}  = - \frac{1}{\rho_0} \nabla P_1 \, ,
</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>
</td></tr>
</table>
</div>

====Wave Equation Derivation====
It is customary to combine these three relations to obtain a single, second-order partial-differential equation in terms of (any) one of the perturbation amplitudes.  We begin by using the third equation to replace <math>~P_1</math> in favor of <math>~\rho_1</math> in the second equation. This generates,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\rho_0 \frac{\partial \vec{v}}{\partial t}  + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, .
</math>
  </td>
</tr>
</table>
</div>
Taking the divergence of this equation gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\rho_0 \frac{\partial}{\partial t}(\nabla\cdot \vec{v})  + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla^2 \rho_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, ;
</math>
  </td>
</tr>
</table>
</div>

while taking the time derivative of the first (''i.e.,'' the linearized continuity) equation gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial^2 \rho_1}{\partial t^2}  + \rho_0 \frac{\partial}{\partial t}(\nabla\cdot \vec{v})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
(Note that we have freely interchanged the order of the <math>~\nabla</math> and <math>~\partial/\partial t</math> operators because the spatial operator is not a function of time.  Also, as before, quantities having a subscript "0" have been pulled outside of both operators because, in this discussion, they have no time- or spatial-dependence.)  Finally, taking the difference between these last two relations produces the anticipated,

<div align="center">
<table border="1" cellpadding="10">
<tr><td align="center">
<font color="#770000">'''Wave Equation'''</font>

<math>~
\frac{\partial^2 \rho_1}{\partial t^2}  - c_s^2 \nabla^2 \rho_1 = 0
</math>
</td></tr>
</table>
</div>
exhibiting the wave propagation speed,
<div align="center">
<math>~
c_s  = \sqrt{\biggl( \frac{dP}{d\rho} \biggr)_0} \, .
</math>
</div>
As derived, this wave equation describes, from an Eulerian (as opposed to Lagrangian) perspective, how the density perturbation, <math>~\rho_1(\vec{r},t)</math>, varies with time at any coordinate position.

====Alternatives====
Utilizing the linearized adiabatic form of the first law of thermodynamics, we can trivially replace <math>~\rho_1</math> with <math>~P_1</math> in the above wave equation to obtain a 
<div align="center">
<table border="0" cellpadding="10">
<tr><td align="center">
<font color="#770000">'''Wave Equation'''</font>

<math>~
\frac{\partial^2 P_1}{\partial t^2}  - c_s^2 \nabla^2 P_1 = 0 \, ,
</math>
</td></tr>
</table>
</div>
that exhibits the same wave propagation speed but describes the variation of the pressure, rather than density, fluctuations.  Alternatively, adopting the approach preferred by [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], we can <FONT COLOR="#007700">introduce the velocity potential by putting</FONT>,
<div align="center">
<math>\vec{v} = \nabla\phi \, ,</math>
</div>
in the linearized governing equations.  After also replacing <math>~P_1</math> in favor of <math>~\rho_1</math> in the linearized Euler equation, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho_0 \frac{\partial \nabla\phi}{\partial t}  + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~ \nabla \biggl[ \rho_0 \frac{\partial \phi}{\partial t}  + \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1 \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~ \frac{\partial \phi}{\partial t}  + \frac{1}{\rho_0}\biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
Taking the time-derivative of this expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\rho_0} \frac{\partial \rho_1}{\partial t} + \biggl( \frac{dP}{d\rho} \biggr)_0^{-1} \frac{\partial^2 \phi}{\partial t^2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0 \, .</math>
  </td>
</tr>
</table>
</div>

In addition, the linearized continuity equation becomes,
<div align="center">
<math>
\frac{1}{\rho_0} \frac{\partial \rho_1}{\partial t}  + \nabla^2 \phi = 0 .
</math>
</div>
Taking the difference between these last two expressions generates a (see, also, equation 63.7 of [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>]),
<div align="center">
<table border="0" cellpadding="10">
<tr><td align="center">
<font color="#770000">'''Wave Equation'''</font>

<math>~
\frac{\partial^2 \phi}{\partial t^2}  - c_s^2 \nabla^2 \phi = 0 \, ,
</math>
</td></tr>
</table>
</div>
that exhibits the same wave propagation speed but describes the variation of the velocity potential, rather than density or pressure fluctuations.