User:Tohline/SSC/SoundWaves

Sound Waves

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, the

Eulerian Representation
of the Continuity Equation,

<math>~\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0</math>

Eulerian Representation
of the Euler Equation,

<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \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, specifically,

<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>

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 begin by investigating small oscillations; an oscillatory motion of small amplitude in a compressible fluid is called a sound wave. Since the oscillations are small, the velocity <math>~\vec{v}</math> is small also, so that the term,

<math>(\vec{v} \cdot \nabla)\vec{v} \, ,</math>

in Euler's equation may be neglected. For the same reason the relative changes in the fluid density and pressure are small. We can write the variables <math>~P</math> and <math>~\rho</math> in the form,

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)

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | 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