Molecular simulations of sound wave propagation in simple gasesNicolas G. Hadjiconstantinou and Alejandro L. Garcia

Citation: Physics of Fluids (1994-present) 13, 1040 (2001); doi: 10.1063/1.1352630 View online: http://dx.doi.org/10.1063/1.1352630 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/13/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Numerical analysis of nonlinear acoustic wave propagation in a rarefied gas AIP Conf. Proc. 1501, 115 (2012); 10.1063/1.4769485 Sound Wave Propagation in Gases at Low Pressure AIP Conf. Proc. 669, 113 (2003); 10.1063/1.1593879 Particle Simulation of Detonation Waves in Rarefied Gases AIP Conf. Proc. 663, 178 (2003); 10.1063/1.1581546 Free molecular sound propagation J. Acoust. Soc. Am. 112, 395 (2002); 10.1121/1.1490360 Sound wave propagation in transition-regime micro- and nanochannels Phys. Fluids 14, 802 (2002); 10.1063/1.1431243

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PHYSICS OF FLUIDS VOLUME 13, NUMBER 4 APRIL 2001

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Molecular simulations of sound wave propagation in simple gasesNicolas G. HadjiconstantinouMechanical Engineering Department, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139

Alejandro L. Garciaa)

Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore,California 94551

~Received 25 July 2000; accepted 11 January 2001!

Molecular simulations of sound waves propagating in a dilute hard sphere gas have been performedusing the direct simulation Monte Carlo method. A wide range of frequencies is investigated,including very high frequencies for which the period is much shorter than the mean collision time.The simulation results are compared to experimental data and approximate solutions of theBoltzmann equation. It is shown that free molecular flow is important at distances smaller than onemean free path from the excitation point. ©2001 American Institute of Physics.@DOI: 10.1063/1.1352630#

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I. INTRODUCTION

In this article we study the propagation of high frquency sound waves in a dilute gas using the direct simtion Monte Carlo~DSMC! technique. Our investigation focuses on the variation of the wave speed and absorpcoefficient as a function of frequency,v. When the period ofoscillation is comparable to the molecular collision time, tsound speed is known to deviate from the classical adiabconstant. Similarly, at high frequencies the attenuationdeviates significantly from the classical behavior observethe low attenuation limit.1,2

Cercignani3 surveys the various molecular kinetic fomulations based on the Boltzmann equation that attempcapture this phenomenon. These theories predict that atfrequencies the sound speed increases with frequencythat the absorption coefficient, which in the classical theincreases proportionally to the square of the frequency,hibits a shallow maximum. Agreement with experimendata4,5 over a frequency range that spans five orders of mnitude varies from moderate to good depending onmodel. Surprisingly, the Maxwell molecule assumptiowhich simplifies the solutions significantly, seems to yiethe same overall level of agreement with experimental das the hard sphere approximation that is more appropriatesimple gases. Among the theoretical approaches discussRef. 3, the 11-moment method of Sirovich and Thurber6 is,in general, in best agreement with the experimental data

We have performed accurate numerical calculationsobtain the wave speed and attenuation coefficient usingdirect simulation Monte Carlo~DSMC! method. These results are a useful complement to the experimental data aable, especially since the latter are subject to significant ster in the high frequency range. Comparison of our reswith analytical solutions can assess the validity of appro

a!Permanent address: Department of Physics, San Jose State UniversitJose, CA 95192-0106. Electronic mail:ngh@mit.edu

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mations involved in the latter; comparison of our results wexperimental data evaluates the ability of the Boltzmaequation as a mathematical model to capture the physreality observed in the experiments. DSMC offers considable flexibility with respect to molecular interaction modethat can be used; the present study uses the hard-spmodel that is typically assumed in the developmenttheory. Further refinements that will improve the connectand direct comparison to experimental data are discussethe last section of the article.

Our calculations confirm that the predictions of thNavier–Stokes, Burnett and super-Burnett theories are oqualitatively correct. Good agreement is found betweenresults and the work of Sirovich and Thurber for hard sphgases. Agreement with experiments is equally good. Thefect of free molecular flow close to the excitation pointalso elucidated through our calculations.

II. WAVE PROPAGATION USING DSMC: NUMERICALTECHNIQUE

Figure 1 shows a sketch of the simulation geometryrectangular domain of lengthL filled with a dilute gas. Stan-dard DSMC7,8 techniques were used to simulate the gas siit is well established that DSMC accurately models hiKnudsen number flows in which the characteristic lengscale~in our case the acoustic wavelength,l! is comparableto the mean free path,l. For the sake of brevity we will notpresent a description of the DSMC algorithm. Excelleintroductory7 and detailed8 descriptions of DSMC, as well aformal derivations,9 can be found in the literature. Comparsons of DSMC simulation results with solutions of the liearized Boltzmann equation and experimental results forverse nonequilibrium phenomena spanning the whKnudsen range can be found in Ref. 8 and 10.

In our simulations sound waves are excited atx50 byimposing a sinusoidally varying velocity in thex-direction.The velocity is imposed using the well-known Maxwellia

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1041Phys. Fluids, Vol. 13, No. 4, April 2001 Molecular simulations of sound waves in simple gases

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reservoir method: particles exiting the domain atx50 arediscarded, whereas particle influx is accounted for by aervoir attached to the simulation domain atx50 and extend-ing to x52LR . Each timestep, particles at the simulatimean density are generated inside the reservoir and are gvelocities drawn from a Maxwellian distribution at the simlation temperature. The Maxwellian distribution has a timdependent mean velocity in thex-direction that is equal tothe desired ‘‘boundary condition.’’ The particle positions aadvanced in time~one timestepDt); the particles that crosthe imaginary planex50 and enter the simulation domarepresent the half-space Maxwellian influx and are retainThe particles remaining in the reservoir are discarded andsimulation proceeds as usual. The length of the reservoset toLR5vcutoffDt, wherevcutoff is a velocity for which theprobability ~based on a Maxwellian distribution! is verysmall. In the simulations we takevcutoff56A2kBT/mm wherekB is Boltzmann’s constant,T is the temperature, andmm isthe molecular mass.

In order to minimize the cost of our simulations the dmain lengthL was taken to be no longer than a few wavlengths. The far end of the domain was terminated bspecular wall leading to total reflection of the propagatwaves. In the absence of dissipation, the system wouldhibit pure standing waves. At low frequencies attenuationrelatively small and the reflected wave is only slightly dimiished with respect to the incoming wave. At higher frequecies no appreciable reflected wave exists.

The simulation geometry resembles that of the acouinterferometer used in the experiments in Ref. 11 but wthree differences. First, the acoustic source is a reserrather than a transducer: the reservoir generates a Maxwian influx distribution which approximates that produceda thermal wall, which in turn approximates the transduused in the experiments; the reservoir, however, doesmodel the change in position of the wall. The second diffence is that experimental measurements are taken at ofew output receivers whereas in the simulations the velovariation,v(x,t), is computed everywhere. This permitsto obtain the sound speed and attenuation coefficient withinterferometry, i.e., without having to vary the domalength. Third, and perhaps most importantly, our evaluatof the velocity field is nonintrusive, in contrast to the receers used in the experiments.

Waves propagating in thex direction lead to a velocityvariationvoexp$i(vt2kx)2mx% wherek52p/ l is the wave-number, andm is the attenuation coefficient. The wave amplitude used was small to avoid nonlinear effects. Speccally, the viscous terms dominate the nonlinear terms w

FIG. 1. Simulation geometry.

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vo!vm/rc wherem is the viscosity,r is the mass densityandc is the classical sound speed.12 It will be convenient toexpress this criterion asvo!c/R where R5c2r/vm is theacoustic Reynolds number.

In the case of high frequency waves (R,0.5) larger ini-tial amplitudes were used~up to vo50.15c) because the attenuation is large and the amplitude damped considera~see Sec. IV!. At intermediate frequencies (0.5<R<4) theamplitude wasvo50.05c, whereas at the lowest frequencie(R.4) the amplitude wasvo50.025c; the above criterion(vo!c/R) is thus not satisfied for R.20. However, the dis-tance that a wave must travel before it significantly steepdue to nonlinear effects12 is Lnl5cl/p(g11)vo , whereg isthe ratio of heat capacities (g5 5

3 for the monatomic gasstudied here!. The systems we consider are small (L5 7

4l ) soLnl>2.7L. From these arguments we expect the effectsnonlinearities to be small at both the high and low frequenranges. Additionally, simulations with the following wavamplitudes were performed:vo50.05c, 0.0125c at R540,vo50.0125c, 0.00625c at R580, andvo50.05c, 0.0125c,0.00625c, 0.003125c at R5120; these confirmed that nonlinear effects were small and did not affect our estimationm andk.

The reflected wave is of the formvoexp$i(vt1kx)2m(2L2x)%. Superposition leads to a standing wave of tform

v~x,t !5vo@~e2mx1e2m~2L2x!!coskx sinvt

2~e2mx2e2m~2L2x!!sinkx cosvt#

5A~x!sinvt1B~x!cosvt. ~1!

After the initial transients have passed, the velocity fieldevaluated at each timestep in slices perpendicular tox-axis. The chi-square fit to this data gives13

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1042 Phys. Fluids, Vol. 13, No. 4, April 2001 N. G. Hadjiconstantinou and A. L. Garcia

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higher frequencies (R,0.5) a phase shift is also includedthe parameter fits. Entrance effects are further discusseSec. IV.

III. THEORIES FOR WAVE PROPAGATION INHOMOGENEOUS MEDIA

The classical Navier–Stokes theory is limited to low frquencies~attenuations! and predicts a constant sound spec5AgkBT/mm, and an attenuation coefficient that is propotional to the square of the frequency2

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At high frequencies, where the attenuation is large,classical result is no longer accurate. The full Navier–Stotheory and its extensions to the Burnett and super-Burlimits have been shown to only qualitatively describe texisting experimental data.4 Despite this, these results auseful because they are given in closed form and perfreasonably well for frequencies that are not too high.0.5). For Maxwell molecules, the complex propagaticonstantb5m/k01 ik/k0 (k05v/c is the classical wavenumber! is given in terms of R as4

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in the super-Burnett theory.Molecular-kinetic calculations have been performed

various levels of approximation and with varying degreessuccess in capturing the variation of the attenuation coecient and sound speed with frequency at the high frequelimit.3 The Boltzmann equation has been solved appromately by Buckner and Ferziger14 using the method of el-ementary solutions and by Sirovich and Thurber6 using themethod of analytic continuation. Both groups consideredlinearized model equation by Gross and Jackson.15 The so-lution by Sirovich and Thurber has been found to perfowell over the whole range for which experimental data ex

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their 8-moment and 11-moment results for hard spheres hbeen transcribed from their paper and are included in Secfor comparison.

IV. SIMULATION RESULTS AND DISCUSSION

We simulated gaseous argon as a hard sphere gas~mo-lecular massmm56.63310226 kg, hard sphere diameters53.66310210 m! at atmospheric conditions (P51.0133105 Pa,T5273 K!. This choice of species was mainly duto historical and convenience reasons. It has been historicused in a large number of DSMC studies because it provinstant availability to a substantial literature of simulatioand experimental results for code validation. The hard sphmodel was selected as the intermolecular potential primabecause it is commonly used in the theoretical analysishigh frequency acoustics. More complicated interactmodels~e.g., variable hard sphere, variable soft sphere! areoften used in DSMC simulations because the temperadependence of the transport coefficients can be importanaccurate computations. However, in our simulations the teperature variations are very small since the wave amplitwas kept small to avoid nonlinear effects.

We ensured that there were 35–40~on average! mol-ecules per cell, which is substantially more than the num~about 20! empirically determined to be required for accurasolutions.16,17 Calculations with approximately 100 moecules per cell at selected frequencies yielded identicalsults. The number of cells was chosen so that the cell lindimension is less than half of a mean free path. The mfree path for hard spheres is given byl5mm /A2ps2r andl'6.2531028 m in our simulations. Alexanderet al.18 haveshown that the transport coefficients deviate from the dilgas Enskog values as the square of the cell sizeDx with theproportionality constant such that for cell sizes of the ordof one mean free path, an error of the order of 10% occIn our caseDx<0.5l, so the error is less than 2.5%. Wused the results of Alexanderet al.18 to correct all our resultsfor the transport coefficients~viscosity and thermal conductivity !.

The timestep of the simulationDt was taken to be sig-nificantly smaller thanl/co where co5A2kBT/mm is themost probable velocity. It has been shown19,20 that the errorin the transport coefficients is proportional to the squarethe timestep, which for accurate simulations has to be msmaller thanl/co . At the highest frequencies the waveforperiod is significantly smaller than the mean free time athus a second constraint on the simulation timestep isDt!2p/v. Consequently, the timestep was set to the highvalue that would satisfy bothDt,l/(5co), and Dt,(2p/v)/60.

Temperature variations due to dissipation were closmonitored and the mean temperature was found to increfor R,1. The maximum temperature increase observedless than 3%, which leads to a change of less than 2% insound speed and transport coefficients given that theseasAT.

Figure 2 shows a comparison between the simulatresults and the classical Navier–Stokes theory forv5422

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1043Phys. Fluids, Vol. 13, No. 4, April 2001 Molecular simulations of sound waves in simple gases

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FIG. 2. Velocity amplitudeV(x)5AA21B2 ~in m/s! asa function of distance~in meters! from the driven endfor v54223106 rad/s (R'20). The solid line repre-sents the Navier–Stokes prediction, the dash-dottedrepresents the simulation result, and the dashedrepresents a fit to the simulation results.

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3106 rad/s. This frequency is low enough that the charteristic time (2p/v;1.531028 s! is much larger than themean collision timet'l/c;1.2310210 s. In this low fre-quency regime the Navier–Stokes theory is well appromated by the classical theory and successfully captuthe physics of wave propagation, as verified by our simution.

Experimentation at high frequencies is difficult becauthe strong absorption makes sampling very close to thenecessary. Greenspan11 limits his experiments to R.0.3 be-cause for R,0.3 the ‘‘wave path is shorter than one mefree path’’ and sufficient signal only exists at distances lthan one mean free path from the input. Measurements clthan one mean free path from the transducer~or from theinput reservoir in the simulations! are likely to be skewed bythe input’s Maxwellian distribution through free moleculflow.6

In his discussion Cercignani argues that sufficient cosions take place even at small distances from the inputsubsequently the region excluded due to free molecular flshould be much smaller than one mean free path. Additally he argues that while molecules with high velocities mbias the distribution function at large distances from theput, this is not expected to happen for hard sphere molecdue to the high collision rate associated with high velochard spheres. We examine these assertions below.

Our results indicate that the wavelength and attenuacoefficient attain constant values forx.l. This is illustratedby considering the same data (R50.5) but restricting the fitto x.xmin with xmin5l in Fig. 3 andxmin50.5l in Fig. 4.Figure 3 shows that the waveforms are fitted very well bconstant sound speed and attenuation coefficient modex.l; the fit extrapolated in the region 0,x,l deterioratesvery quickly, indicating that the free molecular flow is stimportant forx,l. Figure 4 shows that the same consta

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parameters cannot fit the waveform for bothx,l and x.l.

Figures 5 and 6 summarize our computational resultsthe sound speed and the attenuation coefficient in the ra523106 rad/s,v,129.83109 rad/s. Selected experimentadata points from Greenspan,4 and Meyer and Sessler,5 aswell as the theoretical predictions of the Navier–Stokes, Bnett and super-Burnett theories and the 8- and 11-momresults of Sirovich and Thurber are shown. Several ottheories21–23 ~not graphed! are in fair agreement with ousound speed results but predict a smaller absorption cocient at the higher frequencies (R,5).

Unfortunately, for R,0.25 the available signal is verweak and the results obtained usingxmin5l may be inaccu-rate. For R<0.5 we also consideredxmin50.5l and the val-ues ofm andk thus obtained are denoted by the circled stin Figs. 5 and 6. Note that for three frequencies~betweenR50.1 and 0.5! the results from both fits are shown and tvariation in the curve fits indicates the sensitivity of the rsults to the choice ofxmin . As can be seen from Figs. 3 an4, for x,l, that is in the region where free molecular flowsignificant, the wavelength is shorter and the attenuationefficient lower. This is very important in connection with threported experimental results for R,0.25.

We should also reemphasize that for R,4 larger initialamplitudes were gradually introduced due to the appreciaattenuation. The wave amplitude at the reservoir wascreased up tovo /c50.15 to ensure that sufficient signal eisted forx.xmin ~curve fit region!; however, care was taketo ensure thatvoexp(2mxmin)/c,0.025.

In summary, we find that our simulation results aconsistent with the experimental data, and verify the obsvations of Cercignani3 that the classical Navier–Stokedescription fails for R,10, and that none of the proposetheories can capture the variation of both sound sp

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1044 Phys. Fluids, Vol. 13, No. 4, April 2001 N. G. Hadjiconstantinou and A. L. Garcia

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FIG. 3. Cosine and sine componentA(x) andB(x), of the velocity ampli-tude ~in m/s! for v516.23109 rad/s(R50.5). The waveforms are fittedfor x.xmin5l to extract a soundspeed of 464.1 m/s and an attenuatiocoefficient of 1.423107 m21 at thisfrequency. The stars denote simulatiodata used for the fit~solid line! andcrosses denote simulation data not icluded in the fit.

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and attenuation coefficient over the whole frequency ranOur results offer significantly less scatter comparedthe experimental data, which is essential for the developmof an accurate theory valid over the whole frequency ran

V. CONCLUDING REMARKS

In this article we consider the acoustic properties osingle species, dilute gas of hard sphere particles for smamplitude oscillations. This study demonstrates the useness of the simulation technique; extensions to more c

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plex problems directly follow. For example, the sound speand absorption are modified by thermal relaxation effec2

such as vibrational and rotational degrees of freedom, checal reactions, etc. Interesting nonhydrodynamic effects innary mixtures~e.g., ‘‘fast sound’’24,25! are known to occur athigh frequencies. Nonlinear effects become comparableviscous dissipation at amplitudes a few times larger ththose considered here. Finally, the consistent Boltzmalgorithm26 can be employed to simulate nondilute system

Our DSMC technique uses a thermal reservoir with

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FIG. 4. Cosine and sine componentA(x) andB(x), of the velocity ampli-tude ~in m/s! for v516.23109 rad/s(R50.5). The waveforms are fittedfor x.xmin50.5l giving a soundspeed of 515.7 m/s and an attenuatiocoefficient of 1.453107 m21. Thestars denote simulation data used fthe fit ~solid line! and crosses denotesimulation data not included in the fit

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1045Phys. Fluids, Vol. 13, No. 4, April 2001 Molecular simulations of sound waves in simple gases

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FIG. 5. Scaled inverse sound speedscaled inverse frequency. The solilines represent the 8 and 11 momesolutions, the dashed line representhe Navier–Stokes~NS! prediction,the dash-dotted lines represent thBurnett ~B! and super-Burnett~SB!solutions, the crosses represent expemental data (3 for Meyer and Sessler,and 1 for Greenspan!, and the stars(* ) are the DSMC simulation resultsPlain stars denote sampling forxmin

5l and circled stars denote samplinfor xmin50.5l.

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oscillating mean velocity at the domain inlet (x50), to pro-duce the input sound wave. Instead of generatingparticles with a Maxwell–Boltzmann distribution, thChapman–Enskog distribution27 may be used to approximatbulk conditions, possibly minimizing the inhomogeneity asociated with free molecule effects. Alternatively, an osclating solid wall~i.e., a transducer piston! can be used as thacoustic source. This surface should be thermal ratherspecular, otherwise there would be no way to remove

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heat generated by viscous dissipation. Such a wall in cjunction with DSMC collision models that capture real-gnon-hard-sphere interactions8 is expected to be a bettemodel of a typical experimental setup. However, we expthat these different methods of simulating the input acousource will have little effect on the sound speed and atteation prediction presented here since our curve fits excluthe region nearest to the excitation point. However, the phshift in the free molecular regime near the excitation po

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FIG. 6. Scaled absorption coefficienvs scaled inverse frequency. The sollines represent the 8 and 11 momesolutions, the dashed line representhe Navier–Stokes~NS! prediction,the dash-dotted lines represent thBurnett ~B! and super-Burnett~SB!solutions, the crosses represent expemental data (3 for Meyer and Sessler,and 1 for Greenspan!, and the stars(* ) are the DSMC simulation resultsPlain stars denote sampling forxmin

5l and circled stars denote samplinfor xmin50.5l.

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1046 Phys. Fluids, Vol. 13, No. 4, April 2001 N. G. Hadjiconstantinou and A. L. Garcia

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may be sensitive to the various input boundary conditioand an analysis of this effect would be of interest. Finally,replacing the specular surface atx5L with a stationary, con-stant temperature wall, the reflection phase shift of a so2

can be computed.

ACKNOWLEDGMENTS

The authors wish to thank B. Alder and M. MaleMansour for helpful discussions. This work was performwhile one author~N.H.! was visiting the Center for AppliedScientific Computing~CASC! at Lawrence Livermore National Laboratory. The authors would like to thank XabiGaraizar for the hospitality and for making this work posible through the computer resources made available toauthors, and Dr. Kyran Mish, Director, Center for Computional Engineering, Lawrence Livermore National Labortory, for financial support~U.S. Department of EnergyW-7405-ENG-48!. This work was also supported in part bygrant from the European Commission DG12~PSS*1045!.

1J. W. S. Rayleigh,The Theory of Sound~Macmillan, New York, 1896!,Vol. 2.

2L. D. Landau and E. M. Lifshitz,Fluid Mechanics~Pergamon, New York,1989!.

3C. Cercignani,The Boltzmann Equation and its Applications~Springer-Verlag, New York, 1988!.

4M. Greenspan, ‘‘Propagation of sound in five monoatomic gases,’Acoust. Soc. Am.28, 644 ~1956!.

5E. Meyer and G. Sessler, ‘‘Schallausbreitung in Gasen bei hohen Freqzen und sehr niedrigen Drucken,’’ Z. Phys.149, 15 ~1957!.

6L. Sirovich and J. K. Thurber, ‘‘Propagation of forced sound wavesrarefied gasdynamics,’’ J. Acoust. Soc. Am.37, 329 ~1965!.

7F. J. Alexander and A. L. Garcia, ‘‘The direct simulation Monte Camethod,’’ Comput. Phys.11, 588 ~1997!.

8G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of GFlows ~Clarendon, Oxford, 1994!.

9W. Wagner, ‘‘A convergence proof for Bird’s direct simulation MonCarlo method for the Boltzmann equation,’’ J. Stat. Phys.66, 1011~1992!.

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10E. S. Oran, C. K. Oh, and B. Z. Cybyk, ‘‘Direct simulation Monte CarlRecent advances and applications,’’ Annu. Rev. Fluid Mech.30, 403~1998!.

11M. Greenspan and M. C. Thompson, Jr., ‘‘An eleven megacycle interometer for low pressure gases,’’ J. Acoust. Soc. Am.25, 92 ~1953!.

12P. M. Morse and K. U. Ingard,Theoretical Acoustics~McGraw–Hill, NewYork, 1968!.

13W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. FlanneNumerical Recipes in C, 2nd ed. ~Cambridge University Press, Cambridge, 1992!.

14J. Buckner and J. Ferziger, ‘‘Linearized boundary value problem for aand sound propagation,’’ Phys. Fluids9, 2315~1966!.

15E. P. Gross and E. A. Jackson, ‘‘Kinetic models and the linearized Bzmann equation,’’ Phys. Fluids2, 432 ~1959!.

16M. Fallavollita, D. Baganoff, and J. McDonald, ‘‘Reduction of simulatiocost and error for particle simulations of rarefied flows,’’ J. Comput. Ph109, 30 ~1993!.

17G. Chen and I. Boyd, ‘‘Statistical error analysis for the direct simulatiMonte Carlo technique,’’ J. Comput. Phys.126, 434 ~1996!.

18F. Alexander, A. Garcia, and B. Alder, ‘‘Cell size dependence of transpcoefficients in stochastic particle algorithms,’’ Phys. Fluids10, 1540~1998!; erratum,12, 731 ~2000!.

19N. G. Hadjiconstantinou, ‘‘Analysis of discretization in the direct simultion Monte Carlo,’’ Phys. Fluids12, 2634~2000!.

20A. Garcia and W. Wagner, ‘‘Time step truncation error in direct simution Monte Carlo,’’ Phys. Fluids12, 2621~2000!.

21L. C. Woods and H. Troughton, ‘‘Transport processes in dilute gases othe whole range of Knudsen numbers; Part 2, Ultrasonic sound wavesFluid Mech.100, 321 ~1980!.

22A. M. Anile and S. Pluchino, ‘‘Linear wave modes in nonlocal fluid dynamics,’’ Meccanica19, 104 ~1984!.

23G. Lebon and A. Cloot, ‘‘Propagation of ultrasonic sound waves in dispative dilute gases and extended irreversible thermodynamics,’’ WMotion 11, 23 ~1989!.

24R. Huck and E. Johnson, ‘‘Possibility of double sound propagationdisparate-mass gas mixtures,’’ Phys. Rev. Lett.44, 142 ~1980!.

25W. Montfrooij, P. Westerhuijs, V. de Haan, and I. de Schepper, ‘‘Fsound in a helium–neon mixture determined by neutron scattering,’’ PRev. Lett.63, 544 ~1989!.

26F. Alexander, A. Garcia, and B. Alder, ‘‘A consistent Boltzmann algrithm,’’ Phys. Rev. Lett.74, 5212~1995!.

27A. Garcia and B. Alder, ‘‘Generation of the Chapman–Enskog distribtion,’’ J. Comput. Phys.140, 66 ~1998!.

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