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Sound Waves

Whitworth's (1981) Isothermal Free-Energy Surface
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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 [LL75], 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, ~\nabla\Phi 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,

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


Eulerian Representation
of the Euler Equation,

\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P


Adiabatic Form of the
First Law of Thermodynamics

~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0 .

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

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

in which case the adiabatic form of the 1st law of thermodynamics may be written as,


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

This, in turn implies,


\frac{d\ln P}{d\ln\rho} = \gamma_\mathrm{g} \, ,

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

~P = K\rho^{\gamma_\mathrm{g}}    … with …     \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 \, .

Perturbation then Linearization of Equations

Following [LL75] — text in green is taken verbatum from their Chapter VIII (pp. 245-248) — 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, ~\vec{r}, we can write the variables ~P and ~\rho in the form,

~P

~=

~P_0 + P_1(\vec{r},t)  \, ,

~\rho

~=

~\rho_0 + \rho_1(\vec{r},t) \, ,

where ~\rho_0 and ~P_0 are the constant (both in space and time) equilibrium density and pressure, and ~\rho_1 and ~P_1 are their variations in the sound wave ~(|\rho_1/\rho_0 | \ll 1, | P_1/P_0 | \ll 1). Since the oscillations are small — and because we are assuming that the fluid is initially stationary ~(\mathrm{i.e.,}~\vec{v}_0 = 0)the velocity ~\vec{v} is small also. In what follows, by definition, ~P_1, ~\rho_1, and ~\vec{v} 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 ~\rho into the lefthand side of the continuity equation and neglecting small quantities of the second order, we have,

~~\frac{\partial}{\partial t} (\rho_0 + \rho_1) + \nabla\cdot [(\rho_0 + \rho_1)\vec{v}]

~=

~
\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} )}

 

~\approx

~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} \, ,

where, in the first line, the first term on the righthand side has been set to zero because ~\rho_0 is independent of time and, in the second line, ~\rho_0 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

~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v}

~=

~0 \, .

Next, we note that the term,

(\vec{v} \cdot \nabla)\vec{v} \, ,

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

~\frac{1}{(\rho_0 + \rho_1)} \nabla (P_0 + P_1)

~=

~
\frac{1}{\rho_0} \biggl( 1 + \frac{\rho_1}{\rho_0} \biggr)^{-1} \biggl[ \cancelto{0}{\nabla P_0} + \nabla P_1\biggr]

 

~=

~
\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

 

~=

~
\frac{1}{\rho_0} \nabla P_1 - \frac{1}{\rho_0^2} \cancelto{\mathrm{small}}{(\rho_1 \nabla P_1)} \, ,

where, in the first line, ~\nabla P_0 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

~
\frac{\partial \vec{v}}{\partial t}

~=

~
- \frac{1}{\rho_0} \nabla P_1 \, .

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, ~|\vec{v}| \ll c_s.

In a similar fashion, perturbing the variables in the barotropic equation of state leads to,

~
P_0 + P_1

~=

~
K (\rho_0 + \rho_1)^{\gamma_\mathrm{g}}

 

~=

~
K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}}

~\Rightarrow~~~ P_1

~=

~
K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}} - K\rho_0^{\gamma_\mathrm{g}}

 

~=

~
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}}

 

~\approx

~
\gamma_\mathrm{g} \biggl( \frac{P_0}{\rho_0} \biggr) \rho_1 \, .

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

Linearized Equation of State

~P_1

~=

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

Summary

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

Linearized
Equation of Continuity

\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} = 0 ,

Linearized
Euler Equation

~\frac{\partial \vec{v}}{\partial t}  = - \frac{1}{\rho_0} \nabla P_1 \, ,

Linearized
Adiabatic Form of the
First Law of Thermodynamics


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

Wave Equation Derivation

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

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

~=

~
0 \, .

Taking the divergence of this equation gives,

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

~=

~
0 \, ;

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

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

~=

~0 \, .

(Note that we have freely interchanged the order of the ~\nabla and ~\partial/\partial t operators because the spatial operator is not a function of time. Also, as before, quantities having a subscript "0" have been pulled outside of both operators because, in this discussion, they have no time- or spatial-dependence.) Finally, taking the difference between these last two relations produces the anticipated,

Wave Equation

~
\frac{\partial^2 \rho_1}{\partial t^2}  - c_s^2 \nabla^2 \rho_1 = 0

exhibiting the wave propagation speed,

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

As derived, this wave equation describes, from an Eulerian (as opposed to Lagrangian) perspective, how the density perturbation, ~\rho_1(\vec{r},t), varies with time at any coordinate position.

Alternatives

Utilizing the linearized adiabatic form of the first law of thermodynamics, we can trivially replace ~\rho_1 with ~P_1 in the above wave equation to obtain a

Wave Equation

~
\frac{\partial^2 P_1}{\partial t^2}  - c_s^2 \nabla^2 P_1 = 0 \, ,

that exhibits the same wave propagation speed but describes the variation of the pressure, rather than density, fluctuations. Alternatively, adopting the approach preferred by [LL75], we can introduce the velocity potential by putting,

\vec{v} = \nabla\phi \, ,

in the linearized governing equations. After also replacing ~P_1 in favor of ~\rho_1 in the linearized Euler equation, we have,

~\rho_0 \frac{\partial \nabla\phi}{\partial t}  + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_1

~=

~0

~
\Rightarrow ~~~ \nabla \biggl[ \rho_0 \frac{\partial \phi}{\partial t}  + \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1 \biggr]

~=

~0

~
\Rightarrow ~~~ \frac{\partial \phi}{\partial t}  + \frac{1}{\rho_0}\biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1

~=

~0 \, .

Taking the time-derivative of this expression gives,

~\frac{1}{\rho_0} \frac{\partial \rho_1}{\partial t} + \biggl( \frac{dP}{d\rho} \biggr)_0^{-1} \frac{\partial^2 \phi}{\partial t^2}

~=

~ 0 \, .

In addition, the linearized continuity equation becomes,


\frac{1}{\rho_0} \frac{\partial \rho_1}{\partial t}  + \nabla^2 \phi = 0 .

Taking the difference between these last two expressions generates a (see, also, equation 63.7 of [LL75]),

Wave Equation

~
\frac{\partial^2 \phi}{\partial t^2}  - c_s^2 \nabla^2 \phi = 0 \, ,

that exhibits the same wave propagation speed but describes the variation of the velocity potential, rather than density or pressure fluctuations.

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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