User:Tohline/Appendix/Ramblings/SphericalWaveEquation - Revision history

Tohline: /* Second Effort */ Just playing around with 2nd-order ODE, trying to simplify

2016-05-23T03:58:15Z

Second Effort: Just playing around with 2nd-order ODE, trying to simplify

Tohline: /* Assembling the Key Relations */ Finished deriving W-based eigenvalue problem

2016-05-15T02:49:18Z

Assembling the Key Relations: Finished deriving W-based eigenvalue problem

← Older revision		Revision as of 02:49, 15 May 2016
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	+ W \biggl\{ \frac{d^2 \ln(\rho_0 {\bar\sigma}^2)}{dr_0^2}		+ W \biggl\{ \frac{d^2 \ln(\rho_0 {\bar\sigma}^2)}{dr_0^2}
	- \frac{1}{r_0}\biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr]\biggr\} \, .		- \frac{1}{r_0}\biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr]\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Hence, the linearized continuity equation gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~- \biggl(\frac{W\rho_0 {\bar\sigma}^2}{\gamma_gP_0} \biggr)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~r_0 \frac{dx}{dr_0} +3x</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d^2W}{dr_0^2}
			+ \frac{dW}{dr_0} \biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} -\frac{1}{r_0}\biggr]
			+ W \biggl\{ \frac{d^2 \ln(\rho_0 {\bar\sigma}^2)}{dr_0^2}
			- \frac{1}{r_0}\biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr]\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+\frac{3}{r_0}\biggl[
			\frac{dW}{dr_0}
			+ W \cdot\frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0}
			\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d^2W}{dr_0^2}
			+ \frac{dW}{dr_0} \biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} +\frac{2}{r_0}\biggr]
			+ W \biggl\{ \frac{d^2 \ln(\rho_0 {\bar\sigma}^2)}{dr_0^2}
			+ \frac{2}{r_0}\biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr]\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~~ 0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d^2W}{dr_0^2}
			+ \frac{dW}{dr_0} \biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} +\frac{2}{r_0}\biggr]
			+ W \biggl\{ \frac{d^2 \ln(\rho_0 {\bar\sigma}^2)}{dr_0^2}
			+ \frac{2}{r_0}\biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr] + \biggl(\frac{\rho_0 {\bar\sigma}^2}{\gamma_gP_0} \biggr)\biggr\} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Assembling the Key Relations */ Begin second effort

2016-05-15T02:32:04Z

Assembling the Key Relations: Begin second effort

← Older revision		Revision as of 02:32, 15 May 2016
Line 91:		Line 91:
	-->		-->

			===First Effort===
	Let's switch from the perturbation variable, <math>~p</math>, to an enthalpy-related variable,		Let's switch from the perturbation variable, <math>~p</math>, to an enthalpy-related variable,
	<div align="center">		<div align="center">
Line 277:		Line 278:
	<math>~		<math>~
	+\frac{3 (4g_0 + \omega^2 r_0)}{r_0} \biggl[ \frac{dW}{dr_0}+ \frac{W}{\rho_0} \frac{d\rho_0}{dr_0} \biggr]		+\frac{3 (4g_0 + \omega^2 r_0)}{r_0} \biggl[ \frac{dW}{dr_0}+ \frac{W}{\rho_0} \frac{d\rho_0}{dr_0} \biggr]
			</math>
			</td>
			</tr>
			</table>
			</div>



			===Second Effort===
			Let's switch from the perturbation variable, <math>~p</math>, to an enthalpy-related variable,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~W</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{P_1}{\rho_0 {\bar\sigma}^2} = \biggl(\frac{P_0}{\rho_0 {\bar\sigma}^2}\biggr) p \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<math>~{\bar\sigma}^2 \equiv \frac{4g_0}{r_0} + \omega^2 \, .</math>
			</div>

			The second expression then becomes,

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

			<tr>
			<td align="right">
			<math>~xr_0{\bar\sigma}^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{P_0}{\rho_0} \frac{d}{dr_0}\biggl(\frac{W\rho_0 {\bar\sigma}^2}{P_0}\biggr) - \biggl(\frac{g_0 \rho_0 {\bar\sigma}^2}{P_0}\biggr)W</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			{\bar\sigma}^2 \cdot \frac{dW}{dr_0}
			+ W \biggl[ \frac{P_0}{\rho_0} \frac{d}{dr_0}\biggl(\frac{\rho_0 {\bar\sigma}^2}{P_0}\biggr)
			- \biggl(\frac{g_0 \rho_0 {\bar\sigma}^2}{P_0}\biggr)\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~ xr_0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{dW}{dr_0}
			+ \frac{W}{\rho_0{\bar\sigma}^2} \biggl[ P_0 \frac{d}{dr_0}\biggl(\frac{\rho_0 {\bar\sigma}^2}{P_0}\biggr)
			+ \biggl(\frac{\rho_0 {\bar\sigma}^2}{P_0}\biggr)\frac{dP_0}{dr_0} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{dW}{dr_0}
			+ W \biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Taking the derivative of this expression with respect to <math>~r_0</math> gives,

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

			<tr>
			<td align="right">
			<math>~\frac{dx}{dr_0}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d}{dr_0}\biggl\{\frac{1}{r_0}\biggl[ \frac{dW}{dr_0}
			+ W \cdot \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr] \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~~r_0 \frac{dx}{dr_0}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d}{dr_0}\biggl[ \frac{dW}{dr_0}
			+ W \cdot \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr]
			- \frac{1}{r_0}\biggl[ \frac{dW}{dr_0}
			+ W \cdot \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d^2W}{dr_0^2}
			+ \frac{dW}{dr_0} \biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} -\frac{1}{r_0}\biggr]
			+ W \biggl\{ \frac{d^2 \ln(\rho_0 {\bar\sigma}^2)}{dr_0^2}
			- \frac{1}{r_0}\biggl[ \frac{d \ln(\rho_0 {\bar\sigma}^2)}{dr_0} \biggr]\biggr\} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Assembling the Key Relations */ Work toward establishing eigenvalue problem in terms of W

2016-05-15T01:21:14Z

Assembling the Key Relations: Work toward establishing eigenvalue problem in terms of W

Show changes

Tohline: /* Assembling the Key Relations */

2016-05-15T00:03:46Z

Assembling the Key Relations

← Older revision		Revision as of 00:03, 15 May 2016
Line 34:		Line 34:
	</div>		</div>


			<!--
	Let's use the second expression to define the radial perturbation, <math>~x</math>. That is,		Let's use the second expression to define the radial perturbation, <math>~x</math>. That is,
	<div align="center">		<div align="center">
Line 83:		Line 85:
	- \biggl(4g_0 + \omega^2 r_0 \biggr)^{-2} \biggl[\frac{P_0}{\rho_0} \frac{dp}{dr_0} - p g_0\biggr] \frac{d}{dr_0}\biggl(4g_0 + \omega^2 r_0 \biggr)		- \biggl(4g_0 + \omega^2 r_0 \biggr)^{-2} \biggl[\frac{P_0}{\rho_0} \frac{dp}{dr_0} - p g_0\biggr] \frac{d}{dr_0}\biggl(4g_0 + \omega^2 r_0 \biggr)
	</math>		</math>
			</td>
			</tr>
			</table>
			</div>
			-->

			Let's switch from the perturbation variable, <math>~p</math>, to an enthalpy-related variable,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~W</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{P_1}{\rho_0} = \biggl(\frac{P_0}{\rho_0}\biggr)</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Assembling the Key Relations */ Begin working through new p-based wave equation

2016-05-14T20:52:47Z

Assembling the Key Relations: Begin working through new p-based wave equation

← Older revision		Revision as of 20:52, 14 May 2016
Line 34:		Line 34:
	</div>		</div>

			Let's use the second expression to define the radial perturbation, <math>~x</math>. That is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~ x</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{(4g_0 + \omega^2 r_0 ) } \biggl[\frac{P_0}{\rho_0} \frac{dp}{dr_0} - p g_0\biggr] \, .</math>
			</td>
			</tr>
			</table>
			</div>

			Differentiating this relation gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~ \frac{dx}{dr_0}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{(4g_0 + \omega^2 r_0 ) } \frac{d}{dr_0}\biggl[\frac{P_0}{\rho_0} \frac{dp}{dr_0} - p g_0\biggr]
			+ \biggl[\frac{P_0}{\rho_0} \frac{dp}{dr_0} - p g_0\biggr] \frac{d}{dr_0}\biggl(4g_0 + \omega^2 r_0 \biggr)^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{(4g_0 + \omega^2 r_0 ) } \biggl\{ \frac{d}{dr_0}\biggl[\frac{P_0}{\rho_0} \frac{dp}{dr_0}\biggr] - \frac{d}{dr_0}\biggl[p g_0\biggr] \biggr\}
			- \biggl(4g_0 + \omega^2 r_0 \biggr)^{-2} \biggl[\frac{P_0}{\rho_0} \frac{dp}{dr_0} - p g_0\biggr] \frac{d}{dr_0}\biggl(4g_0 + \omega^2 r_0 \biggr)
			</math>
			</td>
			</tr>
			</table>
			</div>

	=See Also=		=See Also=

Tohline: Begin new "ramblings" chapter on radial pulsations

2016-05-14T20:15:25Z

Begin new "ramblings" chapter on radial pulsations

New page

__FORCETOC__ 

=Playing With Spherical Wave Equation=
{{LSU_HBook_header}}

The [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|traditional presentation of the (spherically symmetric) adiabatic wave equation]] focuses on fractional radial displacements, <math>~x \equiv \delta r/r_0</math>, of spherical mass shells. After studying in depth various [[User:Tohline/Apps/PapaloizouPringle84#Nonaxisymmetric_Instability_in_Papaloizou-Pringle_Tori|stability analyses of Papaloizou-Pringle tori]], I have begun to wonder whether the wave equation for spherical polytropes might look simpler if we focus, instead, on fluctuations in the fluid entropy.

==Assembling the Key Relations==
In the traditional approach, the following three linearized equations describe the physical relationship between the three dimensionless perturbation amplitudes <math>~p(r_0) \equiv P_1/P_0</math>, <math>~d(r_0) \equiv \rho_1/\rho_0</math> and <math>~x(r_0) \equiv r_1/r_0</math>, for various characteristic eigenfrequencies, <math>~\omega</math>:

<div align="center">
<table border="1" cellpadding="10">
<tr><td align="center">
'''Linearized''' 
'''Equation of Continuity''' 
<math>
r_0 \frac{dx}{dr_0} = - 3 x - d ,
</math> 

'''Linearized''' 
'''Euler + Poisson Equations''' 
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0} = (4x + p)g_0 + \omega^2 r_0 x ,
</math> 

'''Linearized''' 
Adiabatic Form of the 
'''First Law of Thermodynamics''' 
<math>
p = \gamma_\mathrm{g} d \, .
</math>
</td></tr>
</table>
</div>

=See Also=
* [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Traditional Presentation of Adiabatic Wave Equation]]

{{LSU_HBook_footer}}