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Sound propagation in a relativistic gas mixture
Physica 74 (1974) 330-340
0 North-Holland Publishing Co.
SOUND PROPAGATION
IN A RELATIVISTIC
GAS MIXTURE
J. GUICHELAAR Instituut voor theoretische fysica, Universiteit van Amsterdam, Amsterdam, Nederland
Received 28 November
1973
Synopsis The absorption coefficient of plane-wave disturbances in a relativistic gas mixture of N components is derived, starting from an expression for the entropy production.
1. The characteristic properties of sound e.g., the speed and the absorption can be derived in sundry ways. a) On the macroscopic level one can start from the complete set of equations governing the evolution of the gas, namely the laws, the second law of thermodynamics, irreversible processes and the equations of state. Linearization disturbances lead to the dispersion relation, from which follow the speed of sound and the absorption coefficient immediately1-6). Specifically for the derivation of the absorption coefficient there exists another, more elegant, macroscopic method, based on a consideration
curring in the gas, which means absorption.
irreversible processes ocIn the nonrelativistic
linearization of Boltzmann’s kinetic equation for the distribution function of a gas8sg). In the present paper sound propagation in a gas mixture of N components is studied. The macroscopic method a) will be used to calculate the speed of sound (section 2) and method b) to calculate the absorption coefficient (sections 3, 4, 5). In the last section the connection with nonrelativistic
irreversible processes OCCU~‘~). 330
SOUND
PROPAGATION
IN A RELATIVISTIC
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331
The conservation law of total rest mass reads
De = -@ap,
(1)
where Q is the rest-mass density (measured in the local proper frame of the gas), U’ (,s = 0, 1,2, 3) the hydrodynamic four-velocity of the gas [UP E yv (c, u), yv = (1 - u”/c’)-‘1, 8, the differential operator a/W [X = (ct, x)] and D the substantial time derivative UP a,, . The conservation law for the rest mass of component i (i = 1, . . . , N) can be written as DoI = -g,
ap
(i = 1, . . . . N),
(2)
with pi the rest-mass density of component i (Q = zyzI ei). The conservation of energy and momentum (8,T”” = 0 in terms of the symmetric energy-momentum tensor of a perfect gas: T”” = ~~U”U’C-~ + PA”‘) is given by the following two equations: De + pDe-’
= 0,
(ee + p) c-~DU”
+ AxP allp = 0,
(3)
(4)
in which appear the specific energy e, the hydrostatic pressure p and the projector Ax’ = gx’ + UxUIcc-’ [g’” E diag (- 1, 1, 1, I)]. The second law of thermodynamics reads TDs = De + pDe- 1 - &iDc*¶
(5)
where T is the temperature, s the specific entropy, ,ui the thermodynamic potential and ci the concentration of species i (ci = e:/e). Finally we use an unspecified equation of state, giving the pressure as a functionofe,sandc,(i= l,...,N1): P
=P(e,s,ci).
(6)
Considerable simplification is obtained by observing that Dci = 0
(i = 1, . . ..N).
as follows from (1) and (2). Consequently we have with (3), (5) and (7): Ds = 0.
(7) (8)
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J. GUICHELAAR
Formulae (7) and (8) indicate that the concentrations and the entropy in a small amount of gas do not change in its local proper frame. The connection between the density and pressure changes in the gas is then in view of (6), (7) and (8) DP = @P/W,
c, De.
(9)
We now have three equations left: (l), (4) and (9), relating the changes of Q, U” and p. In order to describe sound propagation we consider small disturbances Q’, p’ and U’” from their equilibrium values eO, pO and Ug = y”, (c, c,). Linearization of eqs. (I), (4) and (9) yields Doe’
= -e.
a,UtP,
(10)
(eoeo + po) c-~D~U’I
+ Or i?g’ = 0,
(10
DOP’ = (ap/aeh, c, Doe’,
(12)
where Do E U,Xd, and Ar = g”” + c-“U,“U~. From these last three equations we obtain for the pressure disturbance p’ the covariant wave equation
0; P!
= Gil (apmh,
c, Ar
8, a,pl,
(13)
where fi is the reduced specific enthalpy .&E (e + pe-l) cW2. Note that e contains the specific rest energy 2. Suppose a solution of this equation is p’ = f(kaxa), describing a disturbance, propagating in the direction k z (k, , k2, k,) with the speed kOc/lkj. For k” we have then the condition ( UlkP)2 = I%-l (ap/a& CiA$“k,,k,. The speed of propagation of the disturbance is then easily seen to be (k parallel to vo) k”c -= Ikl
cs + lvol
1 + Iv01C,C2
;
2_1 cs = f,
ap 0 &-
s,el’
(14)
with c, the speed of sound in the rest frame (v. = 0). 3. The energy of a sound wave. In view of the following calculations we need an expression for the energy of a sound wave. The energy density of the gas is given by the time-time component of the energy-momentum tensor TpY = eeU”U’c-2
+ PA”.
(15)
The total energy E of the fluid is given by the volume integral E=jdVToo=
j dV key:
+ P
01.
(161
SOUND PROPAGATION
333
IN A RELATIVISTIC GAS MIXTURE
In order to obtain from (16) an expression for the sound energy, we expand Too in the global rest frame (u. = 0) up to the second order in the disturbances from equilibrium
Too = eoeo + (eoeo + po) u’~c-~ + (%)S,Cie’
+ + ($)S,Cfe’z.
(17)
The first term on the right-hand side is the equilibrium energy density and does not contribute to the sound energy ES, defined by Es
3
j
dV(Too - eoeo) = E - Eeg.
(18)
The integral of the third term of (17) has to be dealt with carefully. The density variation e’ is measured in the proper frame of the gas and is connected with the global density variation e:, measured in the global rest frame, by the relation e’ = .& - +~ov’2c-2
@ = yileJ,
(19)
in which another second-order term appears. The volume integral over e: vanishes as a result of the conservation of the total rest mass M: (20)
Sdve:=SdVe,-jdVeo=M_M=O. The thermodynamic
see
(> -
ae
s. =,
=
derivatives appearing in (17) can be simply calculated:
c2ho,
( > a2ee
-
W
s. e,
=-.GO
(21)
e0
For the sound energy ES we obtain finally with (17) (18) (19), (20) and (21) :
-5 = j dV13eo&d2 + 3 (&h/eo>e”1.
(22)
By substitution of a plane wave propagating in the x direction with frequency w and wave number K [v: = 0, sin (EC - wt), vi = v:. = 0, e’ = $ sin (Kx - cot)] we can simplify expression (22); using also the relations oeO1$ = KC, [following from (lo)], c, = o/K and j dV sin2 (Kx - cot) = +V we find for the mean soundenergy density in a plane wave ES= ES/V: (23) This expression differs from its nonrelativistic in the nonrelativistic limit).
counterpart
by the factor h (fi + 1
4. The dissipative gas mixture. The energy of a sound wave propagating in a dissipative gas mixture, in which the irreversible processes of viscosity, heat con-
J. GUICHELAAR
334
duction, diffusion and their cross phenomena occur, decreases in the course of time (if no additional sound energy is provided) due to the fact that it is gradually dissipated by the irreversible processes and is transformed into heat. As a result of these processes the entropy of the system increases steadily to a maximum, reached at equilibrium, when the sound wave is completely damped. The connection between the decrease of the sound energy and the entropy increase can be established by the following argument. The sound energy at time t E,(t) can be written as the difference between the total energy E and the energy E&t), which the system would have, if it were brought to thermodynamic equilibrium i.e., with the same entropy [see eq. (18)] : by a reversible “Gedankenexperiment”, E,(t) = E - E&t). Dividing by V we arrive at the corresponding expression for the energy densities B,(t), E and .s,&t) : s,(t) = B - E&t),
(24)
where ~~and B are the mean values averaged over the volume, as a result of which they are not dependent on the space coordinates. Taking the time derivative in the rest frame we have, with the specific energy ee4 defined as e,, z &Jlee,, :
Do%(0 = -eoDoe,, (0,
(25)
because the total energy density B and the equilibrium density e. are constants. Observing that in the equilibrium state at each instant t the concentrations ci are also constants, we can apply the second law to the right-hand side of (25) and find : Doe, = -Q~T~D~s,~.
(26)
The change in the specific entropy s,~ is merely caused by the mean entropy production 0 (per unit volume and time), resulting from the irreversible processes: Doe, = -T,Z.
(27)
This is the essential connection between the sound-energy decrease and the entropy increase. The relaxation time z describing the decrease in the course of time of the soundenergy density [8, N exp (-t/z)] is given by r-l
= jDoe,l/a, = ToZ/$~ohoC;,
(28)
as eqs. (23) and (27) show. 5. The relaxation time and the absorption coefficient. In order to calculate z we shall have to evaluate the mean entropy production 0, as can be seen from eq. (28).
SOUND
PROPAGATION
IN A RELATIVISTIC
GAS MIXTURE
335
In general the entropy production CTcan be written as a sum of terms, each being the product of a flow (e.g. the heat flow) and a thermodynamic force (e.g. the temperature gradient). In relativistic thermodynamics of irreversible processes this expression readslo) : To = JlY,,
+ 5 J;Yi,,
+ Ii?‘Yp,,
(29)
i=1
with the heat flow J,“, the diffusion flows Jr (i = 1, . . . , N) and the viscous flows 17”. The thermodynamic forces Y4,,, Yip and Y,, are given by
Y,,, = -(T-l
a,T + c-‘DU,), (i = 1, . ..) N),
Yip = -Ta,(T-l,u,) ypy =
(30) (31)
-a,u,,
(32)
where the purely relativistic Eckart term c-~DU, potentials /6i (i = 1, . . . , N) in (31). The flows dynamic forces by the phenomenological laws, cients. For the heat flow and the diffusion flows
appears in (30) and the chemical are connected with the thermocontaining the transport coeffiwe havelO*ll)
J,” = LQgApvYqv + ; L,, ApvYlv,
(33)
i=1
Jf = Lig A”‘Ypv + f Lij A”‘Yjye
J=l
(34)
The coefficient Lq4 is related to the heat conduction and Lql, Li, and L,, belong to the diffusion thermo-effect, the thermal diffusion and the ordinary diffusion. Furthermore
we define the traceless tensor lfpy as
For L!“” and the scalar fl.z we have the linear laws:
(37)
with q and vu the coefficients of shear and bulk viscosity, and less symmetric tensor +Ap”AyB (a, U, -I- d, U,) - +Ap” A*# a,U, .
the trace-
336
J. GUICHELAAR
We proceed now by substituting (30)-(37) into (29) :
the flows and the thermodynamic
forces
To = L,,A’” (T-l i&T + c-~DU,,) (T-l a,T + c-~DU,) + 2Td”‘(T-’
a,T + c-~DU,) f L,, 8, (T-l& i=1
+
T2 A”” 5 L,, 8, (T-lpi) 8, (T-‘pj) i.j=l
(38) where we made use of the Onsager relations Li, = Lui. Considering small variations T’, pi and o’ from the equilibrium state, we see that this expression for c is of second order in these variations. Inserting plane waves in the x direction (amplitudes T, /ii and O,, wave number K and frequency co), omitting higher-order terms and taking the mean value of (r over the volume V (giving again a factor +), we arrive at 2TZ = &, (T-lKI? - c-~w~,.)~ + 2(T-lKT
- c-~w&L~~K(/&
- T-$ui~)
i=l
(the indices 0 of the equilibrium quantities T,, and p10 are omitted). The connections between the amplitudes T, GXand ,L$can be taken as in an undamped sound wave (section 2), because we want to have an expression for Eilinear in the transport coefficients. From eq. (10) follows ~6 = eKi?,; moreover we have T = (aT/&&,., 6 and ,i& = (a,uJ&$,,., 8. Consequently we can express T and ,i& in G,: (40) With the help of these relations and c, = co/K we find after some calculation for the mean entropy production (39) :
SOUND
PROPAGATION
IN A RELATIVISTIC
GAS MIXTURE
337
with the purely relativistic quantity y defined as
(42)
For further simplification we can use the thermodynamic
identity
(43)
with c, defined by (14) and the specific heats c,, and c, by
The relaxation time r can now be expressed in terms of the transport coefficients Lg4,L,,,Lij,7 and vu; however, if one considers a source, permanently emitting plane sound waves, one prefers the absorption coefficient oc, describing the attenuation of the amplitude in the direction of propagation. The relaxation time r and the absorption coefficient 01are related by the familiar relation 01 = (2c,z)- l, where the + arises because the energy decreases as the square of the amplitude. The absorption coefficient cxthen becomes with the help of (28), (41) (43) and (44):
cy =
-
-y)”
L,,(l
+ 2(1 - y)T';L,, i=l
=.=I(
(45)
In considering this general formula, we remark that the purely relativistic terms proportional to y and y2 (N cB2 and - c-“) are essentially due to the Eckart term in the heat flow (30). The factor 6-l before the viscosity terms is due to the fact that the expression for the energy contained in the sound wave (23) has an extra factor fi. Formula (45) contains as a special case the result for a one-component system (see for instance Weinberg3)): LX= (w2/2&) [A(1 - JJ)” (JJ - l)/c”r + (4~ + v,)/f;], where the heat conductivity L = L,,/T. The same result was obtained by the author and coworkers, starting from the relativistic version of Boltzmann’s kinetic equationg). For a binary mixture (N = 2) there remain only the coefficients :q,qu,LgQ,Ll, and Lll (as a result of the relationslO) cyZ1 L,,= 0,~~='=, Lij= 0,cJ"=1Lij= 0
338
J. GUICHELAAR
and the Onsager relations Li, = Lai, L,, = Lji). Usually one has the coefficients A, D and D,, the heat conductivity, the diffusion coefficient and the thermaldiffusion coefficient, instead of La*, L,, and LI1. The relations between these coefficients can easily be established. From eqs. (33) and (34) we can derive for the heat flow: J,” = (L,, - L,,LI,L;;)
ApvYqy + LalL1;Jf.
(46)
We define the heat conductivity 1 E (L,, - L,,L1,L;t)/T. This is the proportionality constant between the heat flow and the temperature gradient in the system, when there is no diffusion. For the diffusion flow Jf we can write [(30), (31) and (34)]: A’” avcl - T-l
51” = -L,, (“(“i,
- L IQ
c-~
[L1, - (h, - h,) L,,] A”” a,T
p2)>p,T
A”‘Du
p
- L
(47)
where the partial enthalpy hl = -T2 d (,I_QT-~/~T)~,.~. The usual definitions of D and DT lead to the identifications: D = e-l [a (,ul - ,u2)/ac,],,T LI1, D, clczT. These coefficients 1, D and DT can of course be = substituted in eq. (45) (N = 2) instead of LQQ, L1, and L, 1, but this leads to a much more complicated expression. In the next section some nonrelativistic cases are considered. IL,,
-
@,
-
‘2)
Ll&
6. Nonrelativistic formulae. 6.1. Gases without specified equations of state. a) For an N-component gas mixture the attenuation factor LXcan directly be derived from eq. (45). We note that in the nonrelativistic limit y -0, L + 1 and ,Ui+ C' + pin.*. I'). Because the thermodynamic potentials ,ui contain the specific rest energy of the gas, terms proportional to c2 and c4 appear in eq. (45). However, these terms vanish in view of the relations between the transport coefficients, given in the last paragraph of the preceding section. When we read for all the other quantities in (45) their nonrelativistic counterparts, the expression for 01becomes (omitting the index n.r. of /A~“.~.):
a=-
L,, -I- 2T2fLQ, i=l
s.ct
(48)
SOUND PROPAGATION
IN A RELATIVISTIC GAS MIXTURE
339
b) For a binary mixture the attenuation factor can be found in a paper by Vasu and RCsibois12), it reads
(49)
This result is identical with (48) if N = 2 is taken. The connections between the transport coefficients x, Ly:, LyT and those used in this paper, Lpq, L1, and L, 1, are: TLag = T*x + 2 (h, - h,) LTr + (h, - hJ2 LyF, TL,, = Lyr + (h, - h,) L TF, TL,, = Lyf, with the partial enthalpy hi = -T2 (apiT-‘/8T)p,c,. c) For a one-component gas the damping can be found for instance in the books of Landau and Lifshitz7), and De Groot and Mazur13). 6.2. Ideal gases. I.e., gases with the following equations of state: p = nkT, Iui = kTm;l In Ani (mi/2nkT)3’2, where A is a constant to make the argument dimensionless, k is Boltzmann’s constant, nzI the particle mass, nt the number density of component i and n the total particle density.
a) The formula for the multicomponent and reads
mixture is easily derived from eq. (48)
Note that for an ideal gas (~,u~T-~/~T),,., = 0 and 7” = 0. A formula for a multicomponent system, analogous to (50) may be found in the book of Hirschfelder, Curtiss and Bird6). b) For the binary mixture we can derive from (50) the Kohler formula2): &.=-
2
r3.P 6% 2ec,3
+4r
-
>
1 >
(51)
where the coefficients L, D12 and v12, used by Kohler, are related to Lqg, L,, and L 11 by the relations Til = La4 - L1,Lq,L~~, D12 = kTL,,lnc,c,m,m,, v12 = QC~C~T-~ [L,,Ll: + $kT(m,’ - m;l)]. A derivation of this formula, directly from Boltzmann’s kinetic equations for the gas components, was given by Foch14). c) For the one-component system we recover from (50) the famous Kirchhoff expressionl) by putting Lag/T = il and g/n = m. From a kinetic point of view this case was first worked out by Wang Chang and Uhlenbeck*).
340
J. GUICHELAAR
Acknowledgements. The author wishes to thank Messrs. S.R. de Groot, A. J. Kox and W. A. van Leeuwen for their critical remarks. This investigation is part of the research programme of the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)“, which is’ financially supported by the “Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)“.
REFERENCES 1) Kirchhoff, G., Pogg. Ann. 134 (1868) 177. 2) Kohler, M., Ann. Physik 39 (1941) 209. 3) Weinberg, S., Astrophys. J. 168 (1971) 175. Weinberg, S., Gravitation and Cosmology, John Wiley and Sons, Inc. (New York, 1972). 4) Satb, H., Progr. theor. Phys. 45 (1971) 370. 5) Guichelaar, J., van Leeuwen, W.A. and de Groot, S.R., Physica 59 (1972) 97. 6) Hirschfelder, J.O., Curt&, C.F. and Bird, R.B., Molecular Theory of Gases and Liquids, John Wiley and Sons, Inc. (New York); Chapman and Hall, Ltd. (London, 1954). 7) Landau, L.D. and Lifshitz, E. M., Fluid Mechanics, Pergamon Press (London, 1959). 8) Wang Chang, C.S. and Uhlenbeck, G.E., The Kinetic Theory of Gases; and Foch Jr., J. D. and Ford, G. W., The Dispersion of Sound in Monoatomic Gases, in Studies in Statistical Mechanics V, J. de Boer and G.E.Uhlenbeck, eds., North-Holland Publ. Co. (Amsterdam, 1970). 9) Guichelaar, J., van Leeuwen, W.A. and de Groat, S.R., Physica 68 (1973) 342. 10) Kluitenberg, G.A., de Groot, S.R. and Mazur, P., Physica 19 (1953) 689. 11) de Groat, S.R., van Weert, Ch.G., Hermens, W. Th. and van Leeuwen, W. A., Physica 42 (1969) 309. 12) Vasu, G. and Rksibois, P., Physica 63 (1973) 209. 13) de Groot, S.R. and Mazur, P., Non-equilibrium Thermodynamics, North-Holland Pub]. Co. (Amsterdam, 1962). 14) Foch, J. D., On the Theory of Sound Propagation in Monoatomic Gases and Binary Mixtures, thesis, Rockefeller University, New York (1967).
0 North-Holland Publishing Co.
SOUND PROPAGATION
IN A RELATIVISTIC
GAS MIXTURE
J. GUICHELAAR Instituut voor theoretische fysica, Universiteit van Amsterdam, Amsterdam, Nederland
Received 28 November
1973
Synopsis The absorption coefficient of plane-wave disturbances in a relativistic gas mixture of N components is derived, starting from an expression for the entropy production.
1. The characteristic properties of sound e.g., the speed and the absorption can be derived in sundry ways. a) On the macroscopic level one can start from the complete set of equations governing the evolution of the gas, namely the laws, the second law of thermodynamics, irreversible processes and the equations of state. Linearization disturbances lead to the dispersion relation, from which follow the speed of sound and the absorption coefficient immediately1-6). Specifically for the derivation of the absorption coefficient there exists another, more elegant, macroscopic method, based on a consideration
curring in the gas, which means absorption.
irreversible processes ocIn the nonrelativistic
linearization of Boltzmann’s kinetic equation for the distribution function of a gas8sg). In the present paper sound propagation in a gas mixture of N components is studied. The macroscopic method a) will be used to calculate the speed of sound (section 2) and method b) to calculate the absorption coefficient (sections 3, 4, 5). In the last section the connection with nonrelativistic
irreversible processes OCCU~‘~). 330
SOUND
PROPAGATION
IN A RELATIVISTIC
GAS MIXTURE
331
The conservation law of total rest mass reads
De = -@ap,
(1)
where Q is the rest-mass density (measured in the local proper frame of the gas), U’ (,s = 0, 1,2, 3) the hydrodynamic four-velocity of the gas [UP E yv (c, u), yv = (1 - u”/c’)-‘1, 8, the differential operator a/W [X = (ct, x)] and D the substantial time derivative UP a,, . The conservation law for the rest mass of component i (i = 1, . . . , N) can be written as DoI = -g,
ap
(i = 1, . . . . N),
(2)
with pi the rest-mass density of component i (Q = zyzI ei). The conservation of energy and momentum (8,T”” = 0 in terms of the symmetric energy-momentum tensor of a perfect gas: T”” = ~~U”U’C-~ + PA”‘) is given by the following two equations: De + pDe-’
= 0,
(ee + p) c-~DU”
+ AxP allp = 0,
(3)
(4)
in which appear the specific energy e, the hydrostatic pressure p and the projector Ax’ = gx’ + UxUIcc-’ [g’” E diag (- 1, 1, 1, I)]. The second law of thermodynamics reads TDs = De + pDe- 1 - &iDc*¶
(5)
where T is the temperature, s the specific entropy, ,ui the thermodynamic potential and ci the concentration of species i (ci = e:/e). Finally we use an unspecified equation of state, giving the pressure as a functionofe,sandc,(i= l,...,N1): P
=P(e,s,ci).
(6)
Considerable simplification is obtained by observing that Dci = 0
(i = 1, . . ..N).
as follows from (1) and (2). Consequently we have with (3), (5) and (7): Ds = 0.
(7) (8)
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J. GUICHELAAR
Formulae (7) and (8) indicate that the concentrations and the entropy in a small amount of gas do not change in its local proper frame. The connection between the density and pressure changes in the gas is then in view of (6), (7) and (8) DP = @P/W,
c, De.
(9)
We now have three equations left: (l), (4) and (9), relating the changes of Q, U” and p. In order to describe sound propagation we consider small disturbances Q’, p’ and U’” from their equilibrium values eO, pO and Ug = y”, (c, c,). Linearization of eqs. (I), (4) and (9) yields Doe’
= -e.
a,UtP,
(10)
(eoeo + po) c-~D~U’I
+ Or i?g’ = 0,
(10
DOP’ = (ap/aeh, c, Doe’,
(12)
where Do E U,Xd, and Ar = g”” + c-“U,“U~. From these last three equations we obtain for the pressure disturbance p’ the covariant wave equation
0; P!
= Gil (apmh,
c, Ar
8, a,pl,
(13)
where fi is the reduced specific enthalpy .&E (e + pe-l) cW2. Note that e contains the specific rest energy 2. Suppose a solution of this equation is p’ = f(kaxa), describing a disturbance, propagating in the direction k z (k, , k2, k,) with the speed kOc/lkj. For k” we have then the condition ( UlkP)2 = I%-l (ap/a& CiA$“k,,k,. The speed of propagation of the disturbance is then easily seen to be (k parallel to vo) k”c -= Ikl
cs + lvol
1 + Iv01C,C2
;
2_1 cs = f,
ap 0 &-
s,el’
(14)
with c, the speed of sound in the rest frame (v. = 0). 3. The energy of a sound wave. In view of the following calculations we need an expression for the energy of a sound wave. The energy density of the gas is given by the time-time component of the energy-momentum tensor TpY = eeU”U’c-2
+ PA”.
(15)
The total energy E of the fluid is given by the volume integral E=jdVToo=
j dV key:
+ P
01.
(161
SOUND PROPAGATION
333
IN A RELATIVISTIC GAS MIXTURE
In order to obtain from (16) an expression for the sound energy, we expand Too in the global rest frame (u. = 0) up to the second order in the disturbances from equilibrium
Too = eoeo + (eoeo + po) u’~c-~ + (%)S,Cie’
+ + ($)S,Cfe’z.
(17)
The first term on the right-hand side is the equilibrium energy density and does not contribute to the sound energy ES, defined by Es
3
j
dV(Too - eoeo) = E - Eeg.
(18)
The integral of the third term of (17) has to be dealt with carefully. The density variation e’ is measured in the proper frame of the gas and is connected with the global density variation e:, measured in the global rest frame, by the relation e’ = .& - +~ov’2c-2
@ = yileJ,
(19)
in which another second-order term appears. The volume integral over e: vanishes as a result of the conservation of the total rest mass M: (20)
Sdve:=SdVe,-jdVeo=M_M=O. The thermodynamic
see
(> -
ae
s. =,
=
derivatives appearing in (17) can be simply calculated:
c2ho,
( > a2ee
-
W
s. e,
=-.GO
(21)
e0
For the sound energy ES we obtain finally with (17) (18) (19), (20) and (21) :
-5 = j dV13eo&d2 + 3 (&h/eo>e”1.
(22)
By substitution of a plane wave propagating in the x direction with frequency w and wave number K [v: = 0, sin (EC - wt), vi = v:. = 0, e’ = $ sin (Kx - cot)] we can simplify expression (22); using also the relations oeO1$ = KC, [following from (lo)], c, = o/K and j dV sin2 (Kx - cot) = +V we find for the mean soundenergy density in a plane wave ES= ES/V: (23) This expression differs from its nonrelativistic in the nonrelativistic limit).
counterpart
by the factor h (fi + 1
4. The dissipative gas mixture. The energy of a sound wave propagating in a dissipative gas mixture, in which the irreversible processes of viscosity, heat con-
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334
duction, diffusion and their cross phenomena occur, decreases in the course of time (if no additional sound energy is provided) due to the fact that it is gradually dissipated by the irreversible processes and is transformed into heat. As a result of these processes the entropy of the system increases steadily to a maximum, reached at equilibrium, when the sound wave is completely damped. The connection between the decrease of the sound energy and the entropy increase can be established by the following argument. The sound energy at time t E,(t) can be written as the difference between the total energy E and the energy E&t), which the system would have, if it were brought to thermodynamic equilibrium i.e., with the same entropy [see eq. (18)] : by a reversible “Gedankenexperiment”, E,(t) = E - E&t). Dividing by V we arrive at the corresponding expression for the energy densities B,(t), E and .s,&t) : s,(t) = B - E&t),
(24)
where ~~and B are the mean values averaged over the volume, as a result of which they are not dependent on the space coordinates. Taking the time derivative in the rest frame we have, with the specific energy ee4 defined as e,, z &Jlee,, :
Do%(0 = -eoDoe,, (0,
(25)
because the total energy density B and the equilibrium density e. are constants. Observing that in the equilibrium state at each instant t the concentrations ci are also constants, we can apply the second law to the right-hand side of (25) and find : Doe, = -Q~T~D~s,~.
(26)
The change in the specific entropy s,~ is merely caused by the mean entropy production 0 (per unit volume and time), resulting from the irreversible processes: Doe, = -T,Z.
(27)
This is the essential connection between the sound-energy decrease and the entropy increase. The relaxation time z describing the decrease in the course of time of the soundenergy density [8, N exp (-t/z)] is given by r-l
= jDoe,l/a, = ToZ/$~ohoC;,
(28)
as eqs. (23) and (27) show. 5. The relaxation time and the absorption coefficient. In order to calculate z we shall have to evaluate the mean entropy production 0, as can be seen from eq. (28).
SOUND
PROPAGATION
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335
In general the entropy production CTcan be written as a sum of terms, each being the product of a flow (e.g. the heat flow) and a thermodynamic force (e.g. the temperature gradient). In relativistic thermodynamics of irreversible processes this expression readslo) : To = JlY,,
+ 5 J;Yi,,
+ Ii?‘Yp,,
(29)
i=1
with the heat flow J,“, the diffusion flows Jr (i = 1, . . . , N) and the viscous flows 17”. The thermodynamic forces Y4,,, Yip and Y,, are given by
Y,,, = -(T-l
a,T + c-‘DU,), (i = 1, . ..) N),
Yip = -Ta,(T-l,u,) ypy =
(30) (31)
-a,u,,
(32)
where the purely relativistic Eckart term c-~DU, potentials /6i (i = 1, . . . , N) in (31). The flows dynamic forces by the phenomenological laws, cients. For the heat flow and the diffusion flows
appears in (30) and the chemical are connected with the thermocontaining the transport coeffiwe havelO*ll)
J,” = LQgApvYqv + ; L,, ApvYlv,
(33)
i=1
Jf = Lig A”‘Ypv + f Lij A”‘Yjye
J=l
(34)
The coefficient Lq4 is related to the heat conduction and Lql, Li, and L,, belong to the diffusion thermo-effect, the thermal diffusion and the ordinary diffusion. Furthermore
we define the traceless tensor lfpy as
For L!“” and the scalar fl.z we have the linear laws:
(37)
with q and vu the coefficients of shear and bulk viscosity, and
the trace-
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We proceed now by substituting (30)-(37) into (29) :
the flows and the thermodynamic
forces
To = L,,A’” (T-l i&T + c-~DU,,) (T-l a,T + c-~DU,) + 2Td”‘(T-’
a,T + c-~DU,) f L,, 8, (T-l& i=1
+
T2 A”” 5 L,, 8, (T-lpi) 8, (T-‘pj) i.j=l
(38) where we made use of the Onsager relations Li, = Lui. Considering small variations T’, pi and o’ from the equilibrium state, we see that this expression for c is of second order in these variations. Inserting plane waves in the x direction (amplitudes T, /ii and O,, wave number K and frequency co), omitting higher-order terms and taking the mean value of (r over the volume V (giving again a factor +), we arrive at 2TZ = &, (T-lKI? - c-~w~,.)~ + 2(T-lKT
- c-~w&L~~K(/&
- T-$ui~)
i=l
(the indices 0 of the equilibrium quantities T,, and p10 are omitted). The connections between the amplitudes T, GXand ,L$can be taken as in an undamped sound wave (section 2), because we want to have an expression for Eilinear in the transport coefficients. From eq. (10) follows ~6 = eKi?,; moreover we have T = (aT/&&,., 6 and ,i& = (a,uJ&$,,., 8. Consequently we can express T and ,i& in G,: (40) With the help of these relations and c, = co/K we find after some calculation for the mean entropy production (39) :
SOUND
PROPAGATION
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337
with the purely relativistic quantity y defined as
(42)
For further simplification we can use the thermodynamic
identity
(43)
with c, defined by (14) and the specific heats c,, and c, by
The relaxation time r can now be expressed in terms of the transport coefficients Lg4,L,,,Lij,7 and vu; however, if one considers a source, permanently emitting plane sound waves, one prefers the absorption coefficient oc, describing the attenuation of the amplitude in the direction of propagation. The relaxation time r and the absorption coefficient 01are related by the familiar relation 01 = (2c,z)- l, where the + arises because the energy decreases as the square of the amplitude. The absorption coefficient cxthen becomes with the help of (28), (41) (43) and (44):
cy =
-
-y)”
L,,(l
+ 2(1 - y)T';L,, i=l
=.=I(
(45)
In considering this general formula, we remark that the purely relativistic terms proportional to y and y2 (N cB2 and - c-“) are essentially due to the Eckart term in the heat flow (30). The factor 6-l before the viscosity terms is due to the fact that the expression for the energy contained in the sound wave (23) has an extra factor fi. Formula (45) contains as a special case the result for a one-component system (see for instance Weinberg3)): LX= (w2/2&) [A(1 - JJ)” (JJ - l)/c”r + (4~ + v,)/f;], where the heat conductivity L = L,,/T. The same result was obtained by the author and coworkers, starting from the relativistic version of Boltzmann’s kinetic equationg). For a binary mixture (N = 2) there remain only the coefficients :q,qu,LgQ,Ll, and Lll (as a result of the relationslO) cyZ1 L,,= 0,~~='=, Lij= 0,cJ"=1Lij= 0
338
J. GUICHELAAR
and the Onsager relations Li, = Lai, L,, = Lji). Usually one has the coefficients A, D and D,, the heat conductivity, the diffusion coefficient and the thermaldiffusion coefficient, instead of La*, L,, and LI1. The relations between these coefficients can easily be established. From eqs. (33) and (34) we can derive for the heat flow: J,” = (L,, - L,,LI,L;;)
ApvYqy + LalL1;Jf.
(46)
We define the heat conductivity 1 E (L,, - L,,L1,L;t)/T. This is the proportionality constant between the heat flow and the temperature gradient in the system, when there is no diffusion. For the diffusion flow Jf we can write [(30), (31) and (34)]: A’” avcl - T-l
51” = -L,, (“(“i,
- L IQ
c-~
[L1, - (h, - h,) L,,] A”” a,T
p2)>p,T
A”‘Du
p
- L
(47)
where the partial enthalpy hl = -T2 d (,I_QT-~/~T)~,.~. The usual definitions of D and DT lead to the identifications: D = e-l [a (,ul - ,u2)/ac,],,T LI1, D, clczT. These coefficients 1, D and DT can of course be = substituted in eq. (45) (N = 2) instead of LQQ, L1, and L, 1, but this leads to a much more complicated expression. In the next section some nonrelativistic cases are considered. IL,,
-
@,
-
‘2)
Ll&
6. Nonrelativistic formulae. 6.1. Gases without specified equations of state. a) For an N-component gas mixture the attenuation factor LXcan directly be derived from eq. (45). We note that in the nonrelativistic limit y -0, L + 1 and ,Ui+ C' + pin.*. I'). Because the thermodynamic potentials ,ui contain the specific rest energy of the gas, terms proportional to c2 and c4 appear in eq. (45). However, these terms vanish in view of the relations between the transport coefficients, given in the last paragraph of the preceding section. When we read for all the other quantities in (45) their nonrelativistic counterparts, the expression for 01becomes (omitting the index n.r. of /A~“.~.):
a=-
L,, -I- 2T2fLQ, i=l
s.ct
(48)
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IN A RELATIVISTIC GAS MIXTURE
339
b) For a binary mixture the attenuation factor can be found in a paper by Vasu and RCsibois12), it reads
(49)
This result is identical with (48) if N = 2 is taken. The connections between the transport coefficients x, Ly:, LyT and those used in this paper, Lpq, L1, and L, 1, are: TLag = T*x + 2 (h, - h,) LTr + (h, - hJ2 LyF, TL,, = Lyr + (h, - h,) L TF, TL,, = Lyf, with the partial enthalpy hi = -T2 (apiT-‘/8T)p,c,. c) For a one-component gas the damping can be found for instance in the books of Landau and Lifshitz7), and De Groot and Mazur13). 6.2. Ideal gases. I.e., gases with the following equations of state: p = nkT, Iui = kTm;l In Ani (mi/2nkT)3’2, where A is a constant to make the argument dimensionless, k is Boltzmann’s constant, nzI the particle mass, nt the number density of component i and n the total particle density.
a) The formula for the multicomponent and reads
mixture is easily derived from eq. (48)
Note that for an ideal gas (~,u~T-~/~T),,., = 0 and 7” = 0. A formula for a multicomponent system, analogous to (50) may be found in the book of Hirschfelder, Curtiss and Bird6). b) For the binary mixture we can derive from (50) the Kohler formula2): &.=-
2
r3.P 6% 2ec,3
+4r
-
>
1 >
(51)
where the coefficients L, D12 and v12, used by Kohler, are related to Lqg, L,, and L 11 by the relations Til = La4 - L1,Lq,L~~, D12 = kTL,,lnc,c,m,m,, v12 = QC~C~T-~ [L,,Ll: + $kT(m,’ - m;l)]. A derivation of this formula, directly from Boltzmann’s kinetic equations for the gas components, was given by Foch14). c) For the one-component system we recover from (50) the famous Kirchhoff expressionl) by putting Lag/T = il and g/n = m. From a kinetic point of view this case was first worked out by Wang Chang and Uhlenbeck*).
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J. GUICHELAAR
Acknowledgements. The author wishes to thank Messrs. S.R. de Groot, A. J. Kox and W. A. van Leeuwen for their critical remarks. This investigation is part of the research programme of the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)“, which is’ financially supported by the “Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)“.
REFERENCES 1) Kirchhoff, G., Pogg. Ann. 134 (1868) 177. 2) Kohler, M., Ann. Physik 39 (1941) 209. 3) Weinberg, S., Astrophys. J. 168 (1971) 175. Weinberg, S., Gravitation and Cosmology, John Wiley and Sons, Inc. (New York, 1972). 4) Satb, H., Progr. theor. Phys. 45 (1971) 370. 5) Guichelaar, J., van Leeuwen, W.A. and de Groot, S.R., Physica 59 (1972) 97. 6) Hirschfelder, J.O., Curt&, C.F. and Bird, R.B., Molecular Theory of Gases and Liquids, John Wiley and Sons, Inc. (New York); Chapman and Hall, Ltd. (London, 1954). 7) Landau, L.D. and Lifshitz, E. M., Fluid Mechanics, Pergamon Press (London, 1959). 8) Wang Chang, C.S. and Uhlenbeck, G.E., The Kinetic Theory of Gases; and Foch Jr., J. D. and Ford, G. W., The Dispersion of Sound in Monoatomic Gases, in Studies in Statistical Mechanics V, J. de Boer and G.E.Uhlenbeck, eds., North-Holland Publ. Co. (Amsterdam, 1970). 9) Guichelaar, J., van Leeuwen, W.A. and de Groat, S.R., Physica 68 (1973) 342. 10) Kluitenberg, G.A., de Groot, S.R. and Mazur, P., Physica 19 (1953) 689. 11) de Groat, S.R., van Weert, Ch.G., Hermens, W. Th. and van Leeuwen, W. A., Physica 42 (1969) 309. 12) Vasu, G. and Rksibois, P., Physica 63 (1973) 209. 13) de Groot, S.R. and Mazur, P., Non-equilibrium Thermodynamics, North-Holland Pub]. Co. (Amsterdam, 1962). 14) Foch, J. D., On the Theory of Sound Propagation in Monoatomic Gases and Binary Mixtures, thesis, Rockefeller University, New York (1967).