User:Tohline/SSC/SoundWaves - Revision history

Tohline: /* Sound Waves */ Clean up presentation a bit

2017-03-08T21:06:23Z

Sound Waves: Clean up presentation a bit

← Older revision		Revision as of 21:06, 8 March 2017
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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] — whose ''general'' properties are well documented throughout the mathematics and physical sciences literature — 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 & linearization technique to inhomogeneous and self-gravitating fluids.		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] — whose ''general'' properties are well documented throughout the mathematics and physical sciences literature — 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 & linearization technique to inhomogeneous and self-gravitating fluids.

	In what follows, we borrow heavily from Chapter VIII of [[User:Tohline/Appendix/References#LL75\|~~Landau & Lifshitz (1975)~~]], as it provides an excellent introductory discussion of sound waves.		In what follows, we borrow heavily from Chapter VIII of [<b>[[User:Tohline/Appendix/References#LL75\|<font color="red">LL75</font>]]</b>], as it provides an excellent introductory discussion of sound waves.

	==Assembling the Key Relations==		==Assembling the Key Relations==
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	===Perturbation then Linearization of Equations===		===Perturbation then Linearization of Equations===

	Following ~~Landau and Lifshitz (1975)~~ — text in green is taken ''verbatum'' from Chapter VIII (pp. 245-248) ~~of LL75~~ — 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 {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}} in the form</FONT>,		Following [<b>[[User:Tohline/Appendix/References#LL75\|<font color="red">LL75</font>]]</b>] — text in green is taken ''verbatum'' from their Chapter VIII (pp. 245-248) — 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 {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}} in the form</FONT>,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</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 [[User:Tohline/Appendix/References#LL75\|LL75]]) the,		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>[[User:Tohline/Appendix/References#LL75\|<font color="red">LL75</font>]]</b>]) the,
	<div align="center">		<div align="center">
	<font color="#770000">'''Linearized Continuity Equation'''</font><br />		<font color="#770000">'''Linearized Continuity Equation'''</font><br />
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	</table>		</table>
	</div>		</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 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 [[User:Tohline/Appendix/References#LL75\|LL75]]) the,		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 [[User:Tohline/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>[[User:Tohline/Appendix/References#LL75\|<font color="red">LL75</font>]]</b>]) the,
	<div align="center">		<div align="center">
	<font color="#770000">'''Linearized Euler Equation'''</font><br />		<font color="#770000">'''Linearized Euler Equation'''</font><br />
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	</table>		</table>
	</div>		</div>
	Ultimately, as emphasized in [[User:Tohline/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>.		Ultimately, as emphasized in [<b>[[User:Tohline/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,		In a similar fashion, perturbing the variables in the barotropic equation of state leads to,
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	</table>		</table>
	</div>		</div>
	Hence, we have (see, also, equation 63.4 of [[User:Tohline/Appendix/References#LL75\|LL75]]) the,		Hence, we have (see, also, equation 63.4 of [<b>[[User:Tohline/Appendix/References#LL75\|<font color="red">LL75</font>]]</b>]) the,
	<div align="center">		<div align="center">
	<font color="#770000">'''Linearized Equation of State'''</font><br />		<font color="#770000">'''Linearized Equation of State'''</font><br />
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	</table>		</table>
	</div>		</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 [[User:Tohline/Appendix/References#LL75\|LL75]], we can <FONT COLOR="#007700">introduce the velocity potential by putting</FONT>,		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>[[User:Tohline/Appendix/References#LL75\|<font color="red">LL75</font>]]</b>], we can <FONT COLOR="#007700">introduce the velocity potential by putting</FONT>,
	<div align="center">		<div align="center">
	<math>\vec{v} = \nabla\phi \, ,</math>		<math>\vec{v} = \nabla\phi \, ,</math>
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	</math>		</math>
	</div>		</div>
	Taking the difference between these last two expressions generates a (see, also, equation 63.7 of [[User:Tohline/Appendix/References#LL75\|LL75]]),		Taking the difference between these last two expressions generates a (see, also, equation 63.7 of [<b>[[User:Tohline/Appendix/References#LL75\|<font color="red">LL75</font>]]</b>]),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="10">		<table border="0" cellpadding="10">

Tohline at 18:07, 8 December 2014

2014-12-08T18:07:06Z

← Older revision		Revision as of 18:07, 8 December 2014
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	</math>		</math>
	</div>		</div>
	Taking the difference between these last two expressions generates a,		Taking the difference between these last two expressions generates a (see, also, equation 63.7 of [[User:Tohline/Appendix/References#LL75\|LL75]]),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="10">		<table border="0" cellpadding="10">

Tohline: /* Summary */ Finished deriving two alternative wave equations.

2014-12-08T18:03:31Z

Summary: Finished deriving two alternative wave equations.

← Older revision		Revision as of 18:03, 8 December 2014
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	</div>		</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,		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">		<div align="center">
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	</math>		</math>
	</div>		</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 [[User:Tohline/Appendix/References#LL75\|LL75]], 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,
			<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.

	=See Also=		=See Also=

Tohline: /* Sound Waves */ Clean up presentation before adding additional text

2014-12-08T17:12:37Z

Sound Waves: Clean up presentation before adding additional text

← Older revision		Revision as of 17:12, 8 December 2014
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	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	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 time-dependent behavior of the variables that simultaneously ~~satisfies~~ 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] — whose ''general'' properties are well documented throughout the mathematics and physical sciences literature — 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 & linearization technique to inhomogeneous and self-gravitating fluids.		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] — whose ''general'' properties are well documented throughout the mathematics and physical sciences literature — 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 & linearization technique to inhomogeneous and self-gravitating fluids.

	In what follows, we borrow heavily from Chapter VIII of [[User:Tohline/Appendix/References#LL75\|Landau & Lifshitz (1975)]], as it provides an excellent introductory discussion of sound waves.		In what follows, we borrow heavily from Chapter VIII of [[User:Tohline/Appendix/References#LL75\|Landau & Lifshitz (1975)]], as it provides an excellent introductory discussion of sound waves.
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	Ultimately, as emphasized in [[User:Tohline/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>.		Ultimately, as emphasized in [[User:Tohline/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>.

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

	It is customary to combine these three relations to obtain a single, second-order ~~ODE~~ 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,		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">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	Finally, taking the difference between these two relations produces the anticipated,		(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">		<div align="center">

Tohline: /* Summary */ Highlight derived wave equation

2014-12-08T02:56:15Z

Summary: Highlight derived wave equation

← Older revision		Revision as of 02:56, 8 December 2014
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	</table>		</table>
	</div>		</div>
	Finally, taking the difference between these two relations ~~gives~~,		Finally, taking the difference between these two relations produces the anticipated,

	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="1" cellpadding="10">
			<tr><td align="center">
			<font color="#770000">'''Wave Equation'''</font>

	~~<tr>~~
	~~<td align="right">~~
	<math>~		<math>~
	\frac{\partial^2 \rho_1}{\partial t^2} - c_s^2 \nabla^2 \rho_1		\frac{\partial^2 \rho_1}{\partial t^2} - c_s^2 \nabla^2 \rho_1 = 0
	</math>		</math>
	</td>		</td></tr>
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~0 \, ,</math>~~
	~~</td>~~
	</tr>
	</table>		</table>
	</div>		</div>
			exhibiting the wave propagation speed,
	~~which is~~ the ~~anticipated ''wave equation'' exhibiting a~~ wave propagation speed,
	<div align="center">		<div align="center">
	<math>~		<math>~

Tohline: /* Assembling the Key Relations */ Finish deriving the wave equation

2014-12-08T02:47:57Z

Assembling the Key Relations: Finish deriving the wave equation

← Older revision		Revision as of 02:47, 8 December 2014
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	</tr>		</tr>
	</table>		</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>

			It is customary to combine these three relations to obtain a single, second-order ODE 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>
			Finally, taking the difference between these two relations gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~
			\frac{\partial^2 \rho_1}{\partial t^2} - c_s^2 \nabla^2 \rho_1
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 \, ,</math>
			</td>
			</tr>
			</table>
			</div>

			which is the anticipated ''wave equation'' exhibiting a wave propagation speed,
			<div align="center">
			<math>~
			c_s = \sqrt{\biggl( \frac{dP}{d\rho} \biggr)_0} \, .
			</math>
	</div>		</div>

Tohline: /* Perturbation then Linearization of Equations */

2014-12-08T01:57:47Z

Perturbation then Linearization of Equations

← Older revision		Revision as of 01:57, 8 December 2014
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	<math>~		<math>~
	\gamma_\mathrm{g} \biggl( \frac{P_0}{\rho_0} \biggr) \rho_1 \, .		\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 [[User:Tohline/Appendix/References#LL75\|LL75]]) 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>		</math>
	</td>		</td>

Tohline: /* Governing Equations and Supplemental Relations */

2014-12-08T01:51:05Z

Governing Equations and Supplemental Relations

← Older revision		Revision as of 01:51, 8 December 2014
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	<div align="center">		<div align="center">

	<math>~P = K\rho^{\gamma_\mathrm{g}}</math>    … with …     <math>\gamma_\mathrm{g} \equiv \frac{d\ln P_0}{d\ln \rho_0} \, .</math>		<math>~P = K\rho^{\gamma_\mathrm{g}}</math>    … with …
			<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>		</div>

Tohline: /* Assembling the Key Relations */ Linearized equation of state

2014-12-08T01:47:19Z

Assembling the Key Relations: Linearized equation of state

← Older revision		Revision as of 01:47, 8 December 2014
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	</div>		</div>
	Ultimately, as emphasized in [[User:Tohline/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>.		Ultimately, as emphasized in [[User:Tohline/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>.

			Finally, 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">

			</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">

			</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">

			</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>

	=See Also=		=See Also=

Tohline: /* Governing Equations and Supplemental Relations */ Insert barotropic equation of state

2014-12-08T01:21:03Z

Governing Equations and Supplemental Relations: Insert barotropic equation of state

← Older revision		Revision as of 01:21, 8 December 2014
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	<div align="center">		<div align="center">
	<math>		<math>
	\rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0		\rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 \, .
	</math>		</math>
			</div>

			This, in turn implies,
			<div align="center">
	<math>		<math>
	~~\Rightarrow ~~~~~ \frac{d\ln P}{d\ln\rho} = \gamma_\mathrm{g} \, .		\frac{d\ln P}{d\ln\rho} = \gamma_\mathrm{g} \, ,
	</math>		</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>    … with …     <math>\gamma_\mathrm{g} \equiv \frac{d\ln P_0}{d\ln \rho_0} \, .</math>
	</div>		</div>