Sound Waves

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 time-dependent behavior of the variables that simultaneously satisfies 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.

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 drop the Poisson equation altogether. Specifically, the relevant set of governing equations is, 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> .

We supplement this set of equations with an ideal gas equation of state, specifically,

<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math> ,

in which case the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics may be written as,

<math> \rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 \, . </math>

This, in turn implies,

<math> \frac{d\ln P}{d\ln\rho} = \gamma_\mathrm{g} \, , </math>

which we will enforce by adopting the barotropic (polytropic) equation of state,

<math>~P = K\rho^{\gamma_\mathrm{g}}</math> … with … <math>\gamma_\mathrm{g} \equiv \frac{d\ln P_0}{d\ln \rho_0} = \frac{\rho_0}{P_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \, .</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. Given that the relative changes in the fluid density and pressure are small, in this Eulerian analysis where we are investigating how conditions vary with time at a fixed point in space, <math>~\vec{r}</math>, we can write the variables <math>~P</math> and <math>~\rho</math> in the form,

<math>~P</math>	<math>~=</math>	<math>~P_0 + P_1(\vec{r},t) \, ,</math>
<math>~\rho</math>	<math>~=</math>	<math>~\rho_0 + \rho_1(\vec{r},t) \, ,</math>

where <math>~\rho_0</math> and <math>~P_0</math> are the constant (both in space and time) equilibrium density and pressure, and <math>~\rho_1</math> and <math>~P_1</math> are their variations in the sound wave <math>~(|\rho_1/\rho_0 | \ll 1, | P_1/P_0 | \ll 1)</math>. Since the oscillations are small — and because we are assuming that the fluid is initially stationary <math>~(\mathrm{i.e.,}~\vec{v}_0 = 0)</math> — the velocity <math>~\vec{v}</math> is small also. In what follows, by definition, <math>~P_1</math>, <math>~\rho_1</math>, and <math>~\vec{v}</math> are considered to be of first order in smallness, while products of these quantities are of second (or even higher) order in smallness.

Substituting the expression for <math>~\rho</math> into the lefthand side of the continuity equation and neglecting small quantities of the second order, we have,

<math>~~\frac{\partial}{\partial t} (\rho_0 + \rho_1) + \nabla\cdot [(\rho_0 + \rho_1)\vec{v}]</math>	<math>~=</math>	<math>~ \cancelto{0}{\frac{\partial \rho_0}{\partial t}} + \frac{\partial \rho_1}{\partial t} + \nabla\cdot (\rho_0 \vec{v}) + \nabla\cdot\cancelto{\mathrm{small}}{(\rho_1\vec{v} )} </math>
	<math>~\approx</math>	<math>~ \frac{\partial \rho_1}{\partial t} + \rho_0\nabla\cdot \vec{v} \, , </math>

where, in the first line, the first term on the righthand side has been set to zero because <math>~\rho_0</math> is independent of time and, in the second line, <math>~\rho_0</math> has been pulled outside of the divergence operator because we have assumed that the initial equilibrium state is homogeneous. Hence, we have (see, also, equation 63.2 of LL75) the,

Linearized Continuity Equation

<math>~ \frac{\partial \rho_1}{\partial t} + \rho_0\nabla\cdot \vec{v} </math>

Next, we note that the term,

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

in Euler's equation may be neglected because it is of second order in smallness. Substituting the expressions for <math>~\rho</math> and <math>~P</math> into the righthand side of the Euler equation and neglecting small quantities of the second order, we have,

<math>~\frac{1}{(\rho_0 + \rho_1)} \nabla (P_0 + P_1)</math>	<math>~=</math>	<math>~ \frac{1}{\rho_0} \biggl( 1 + \frac{\rho_1}{\rho_0} \biggr)^{-1} \biggl[ \cancelto{0}{\nabla P_0} + \nabla P_1\biggr] </math>
	<math>~=</math>	<math>~ \frac{1}{\rho_0} \biggl[ 1 - \frac{\rho_1}{\rho_0} + \cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2} + \cdots \biggr] \nabla P_1 </math>
	<math>~=</math>	<math>~ \frac{1}{\rho_0} \nabla P_1 - \frac{1}{\rho_0^2} \cancelto{\mathrm{small}}{(\rho_1 \nabla P_1)} \, , </math>

where, in the first line, <math>~\nabla P_0</math> has been set to zero because we have assumed that the initial equilibrium state is homogeneous, and the binomial theorem has been used to obtain the expression on the righthand side of the second line. Combining these simplification steps, we have (see, also, equation 63.3 of LL75) the,

Linearized Euler Equation

<math>~ \frac{\partial \vec{v}}{\partial t} </math>

<math>~ - \frac{1}{\rho_0} \nabla P_1 \, . </math>

Ultimately, as emphasized in LL75, the condition that the linearized governing equations should be applicable to the propagation of sound waves is that the velocity of the fluid particles in the wave should be small compared with the velocity of sound, that is, <math>~|\vec{v}| \ll c_s</math>.

Finally, perturbing the variables in the barotropic equation of state leads to,

<math>~ P_0 + P_1 </math>	<math>~=</math>	<math>~ K (\rho_0 + \rho_1)^{\gamma_\mathrm{g}} </math>
	<math>~=</math>	<math>~ K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}} </math>
<math>~\Rightarrow~~~ P_1</math>	<math>~=</math>	<math>~ K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}} - K\rho_0^{\gamma_\mathrm{g}} </math>
	<math>~=</math>	<math>~ K\rho_0^{\gamma_\mathrm{g}} \biggl[1 + \gamma_\mathrm{g}\biggl(\frac{\rho_1}{\rho_0} \biggr) + \frac{\gamma_\mathrm{g}(\gamma_\mathrm{g}-1)}{2} \cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2} + \cdots \biggr] - K\rho_0^{\gamma_\mathrm{g}} </math>
	<math>~\approx</math>	<math>~ \gamma_\mathrm{g} \biggl( \frac{P_0}{\rho_0} \biggr) \rho_1 \, . </math>

Hence, we have (see, also, equation 63.4 of LL75) the,

Linearized Equation of State

<math>~ \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\, . </math>

Summary

In summary, the following three linearized equations govern the time-dependent physical relationship between the three perturbation amplitudes <math>~P_1(\vec{r},t)</math>, <math>~\rho_1(\vec{r},t)</math> and <math>~\vec{v}(\vec{r},t)</math> in the context of sound waves:

Linearized
Equation of Continuity
<math> \frac{\partial \rho_1}{\partial t} + \rho_0\nabla\cdot \vec{v} = 0 , </math>

Linearized
Euler Equation
<math> ~\frac{\partial \vec{v}}{\partial t} = - \frac{1}{\rho_0} \nabla P_1 \, , </math>

Linearized
Adiabatic Form of the
First Law of Thermodynamics

<math> P_1 = \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\, . </math>

It is customary to combine these three relations to obtain a single, second-order ODE in terms of (any) one of the perturbation amplitudes. We begin by using the third equation to replace <math>~P_1</math> in favor of <math>~\rho_1</math> in the second equation. This generates,

<math>~ \rho_0 \frac{\partial \vec{v}}{\partial t} + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_1 </math>

Taking the divergence of this equation gives,

<math>~ \rho_0 \frac{\partial}{\partial t}(\nabla\cdot \vec{v}) + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla^2 \rho_1 </math>

while taking the time derivative of the first (i.e., the linearized continuity) equation gives,

<math>~ \frac{\partial^2 \rho_1}{\partial t^2} + \rho_0 \frac{\partial}{\partial t}(\nabla\cdot \vec{v}) </math>

Finally, taking the difference between these two relations produces the anticipated,

Wave Equation

<math>~ \frac{\partial^2 \rho_1}{\partial t^2} - c_s^2 \nabla^2 \rho_1 = 0 </math>

exhibiting the wave propagation speed,

<math>~ c_s = \sqrt{\biggl( \frac{dP}{d\rho} \biggr)_0} \, . </math>

User:Tohline/SSC/SoundWaves

Contents

Sound Waves

Assembling the Key Relations

Governing Equations and Supplemental Relations

Perturbation then Linearization of Equations

Summary

See Also

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