User:Tohline/SSC/SoundWaves
Sound Waves
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A standard technique that is used throughout astrophysics to test the stability of self-gravitating fluids involves perturbing then linearizing each of the principal governing equations before seeking time-dependent solutions 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.
The discussion of sound waves provided in Chapter VIII of Landau & Lifshitz (1975) remains one of the best, so we will borrow heavily from it.
Assembling the Key Relations
Governing Equations and Supplemental Relations
We begin with the set of 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 ignore the Poisson equation altogether. The set of governing equations is, therefore,
Equation of Continuity
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<math>\frac{d\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math> |
Euler Equation
<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P </math>
Adiabatic Form of the
First Law of Thermodynamics
<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math> ,
supplemented by an ideal gas equation of state and, specifically, the relation,
<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>.
As a result, the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics can be written as,
<math> \rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 </math>
<math> \Rightarrow ~~~ \frac{d\ln P}{d\ln\rho} = \gamma_\mathrm{g} \, . </math>
See Also
- Part II of Spherically Symmetric Configurations: Stability
- Wave Equation
- Sound Waves and Gravitational Instability — class notes provided online by David H. Weinberg (The Ohio State University)
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