User:Tohline/Appendix/Ramblings/Azimuthal Distortions
Analyzing Azimuthal Distortions
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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 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 Landau & Lifshitz (1975), as it provides an excellent introductory discussion of sound waves.
Adopted Notation
We will adopt the notation of K. Hadley & J. N. Imamura (2011a, Astrophysics and Space Science, 334, 1). Specifically, drawing on their equation (6) but ignoring variations in the vertical coordinate, the mass density is given by the expression,
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<math>~\rho</math> |
<math>~=</math> |
<math>~\rho_0 + \delta\rho(\varpi,t)e^{im\phi} \, .</math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |