User:Tohline/Appendix/Ramblings/SphericalWaveEquation

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Playing With Spherical Wave Equation

Whitworth's (1981) Isothermal Free-Energy Surface
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The 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 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>:

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>


Let's switch from the perturbation variable, <math>~p</math>, to an enthalpy-related variable,

<math>~W</math>

<math>~\equiv</math>

<math>~\frac{P_1}{\rho_0} = \biggl(\frac{P_0}{\rho_0}\biggr)</math>

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation