Velocity and Absorption Coefficient of Sound Waves in Classical Gases
Keywords:hydrodynamics, kinetic approach, ultrasonic plane sound waves, velocity, absorption
The velocity and absorption coefficient of plane sound waves in classical gases are obtained by solving the Boltzmann kinetic equation. This is done within the linear response theory as a reaction of the single-particle distribution function to a periodic external field. The nonperturbative dispersion equation is derived in the relaxation time approximation and solved numerically. The obtained theoretical results demonstrate an universal dependence of the sound velocity and scaled absorption coefficient on the variable wт , where w is the sound frequency, and т−1 is the particle collision frequency. In the region of wт ∼ 1, a transition from the frequent- to rare-collision regime takes place. The sound velocity increases sharply, and the scaled absorption coefficient has a maximum – both theoretical findings are in agreement with the data.
L.D. Landau, E.M. Lifshitz. Fluid Mechanics (Pergamon Press, 1987).
K.B. Tolpygo. Thermodynamics and Statistical Physics (Kiev University, 1966) (in Russian).
J.H. Fertziger, H.G. Kaper, Mathematical Theory of Transport Processes in Gases (North-Holand, 1972).
S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge Univ. Press, 1990).
E.M. Lifshitz, L.P. Pitaevskii. Physical Kinetics (Elsevier, 1981).
L. D. Landau, E. M. Lifshitz. Statistical Physics (Pergamon, 1980).
C. Cercignani. Mathematical Methods in Kinetic Theory (Springer, 1969). https://doi.org/10.1007/978-1-4899-5409-1
C. Cercignani. Theory and Application of the Boltzmann Equation (Scottish Academic Press, 1975).
M. Greenspan. Propagation of sound in five monatomic gases. J. Acoust. Soc. Am. 28, 644 (1956). https://doi.org/10.1121/1.1908432
M. Greenspan. Transmission of sound waves in gases at very low pressures. In: Physical Acoustics, edited by W.P. Mason (Academic, 1965), Vol. II, Part A. https://doi.org/10.1016/B978-1-4832-2858-7.50009-0
R. Schotter. Rarefied gas acoustics in the noble gases. Phys. Fluids 17, 1163 (1974). https://doi.org/10.1063/1.1694859
M.E. Meyer, G. Sessler. Schallausbreitung in Gasen bei hohen Frequenzen und sehr niedrigen Drucken. Z. Phys. 149, 15 (1957). https://doi.org/10.1007/BF01325690
C.S. Wang Chang. On the Dispersion of Sound in Helium. Report APL/JHU CM-467, UMH-3-F (University of Michigan, 1948).
C.S. Wang Chang, G.E. Uhlenbeck. The heat transport between two parallel plates as function of the Knudsen number. In: Studies in Statistical Mechanics. Edited by J. de Boer and G.E. Uhlenbeck (Elsevier, 1970), Vol. V, pp. 43-75.
L.C. Woods, H. Troughton. Transport processes in dilute gases over the whole range of Knudsen numbers. Part 2. Ultrasonic sound waves. J. Fluid. Mech. 100, 321 (1980). https://doi.org/10.1017/S0022112080001176
G. Lebon, A. Cloot. Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics. Wave Motion 11, 23 (1989). https://doi.org/10.1016/0165-2125(89)90010-3
M.N. Kogan, Dynamics of the Dilute Gas. Kinetic Theory (Nauka, 1967) (in Russian).
P.L. Bhatnagar, E.P. Gross, M. Krook. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511 (1954). https://doi.org/10.1103/PhysRev.94.511
E.P. Gross, E.A. Jackson. Kinetic models and the linearized Boltzmann equation. Phys. Fluids 2, 432 (1959). https://doi.org/10.1063/1.1724415
H. Heiselberg, C.J. Pethick, D.G. Revenhall. Collective behavior of stable and unstable hot Fermi systems. Ann. Phys. (N.Y.) 223, 37 (1993). https://doi.org/10.1006/aphy.1993.1026
M. Takamoto, S. Inutsuka. The relativistic kinetic dispersion relation: Comparison of the relativistic Bhatnagar-Gross-Krook model and Grad's 14-moment expansion. Prog. Theor. Phys. 123, 903 (2010). https://doi.org/10.1016/j.physa.2010.06.021
A.G. Magner, D.V. Gorpinchenko, J. Bartel. A semiclassical collective response of heated, asymmetric, and rotating nuclei. Phys. At. Nucl. 77, 1229 (2014). https://doi.org/10.1134/S1063778814090051
L. Sirovich, J.K. Thurber. Propagation of forced sound waves in rarefied gasdynamics. J. Acoust. Soc. Am. 37, 329 (1965). https://doi.org/10.1121/1.1909331
J.K. Buckner, J.H. Ferziger. Linearized boundary value problem for a gas and sound propagation. Phys. Fluids 9, 2315 (1966). https://doi.org/10.1063/1.1761620
J.R. Thomas Jr., C.E. Siewert. Sound-wave propagation in a rarefied gas. Transp. Theory Statist. Phys. 8, 219 (1979). https://doi.org/10.1080/00411457908214538
S.K. Loyalka, T.C. Cheng. Sound-wave propagation in a rarefied gas. Phys. Fluids 22, 830 (1979). https://doi.org/10.1063/1.862669
T.C. Cheng, S.K. Loyalka. Sound wave propagation in a rarefied gas-II: Gross-Jackson model. Prog. Nucl. Energy 8, 263 (1981). https://doi.org/10.1016/0149-1970(81)90020-2
R.D.M. Garcia, C.E. Siewert. The linearized Boltzmann equation: Sound-wave propagation in a rarefied gas. Z. angew. Math. Phys. 57, 94 (2005). https://doi.org/10.1007/s00033-005-0007-8
F. Sharipov, D. Kalempa. Numerical modeling of the sound propagation through a rarefied gas in a semi-infinite space on the basis of linearized kinetic equation. J. Acoust. Soc. Am. 124, 1993 (2008). https://doi.org/10.1121/1.2967835
I.L. Bekharevich, I.M. Khalatnikov. Theory of Kapitsa heat jump on a solid boundary. Sov. J. Exp. Theor. Phys. 12, 1187 (1961).
Yu.B. Ivanov. Fermi-liquid surface vibrations. Nucl. Phys. A 365, 301 (1981). https://doi.org/10.1016/0375-9474(81)90298-0
A.G. Magner. Boundary conditions for semiclassical distribution function. Sov. J. Nucl. Phys. 45, 235 (1987).
N.G. Hadjiconstantinou, A.L. Garcia. Molecular simulations of sound wave propagation in simple gases. Phys. of Fluids 13, 1040 (2001). https://doi.org/10.1063/1.1352630
V.M. Strutinsky, S.M. Vydrug-Vlasenko, A.G. Magner. Macroscopic resonances in planar geometry. Z. Phys. A 327, 267 (1987). https://doi.org/10.1007/BF01284449
V.M. Kolomietz, A.G. Magner, V.M. Strutinsky, S.M. Vydrug-Vlasenko. Monopole modes in a finite Fermi system with diffuse reflection boundary conditions. Nucl. Phys. A 571, 117 (1994). https://doi.org/10.1016/0375-9474(94)90344-1
A.G. Magner, M.I. Gorenstein, U.V. Grygoriev. Viscosity of a classical gas: The rare-collision versus the frequent-collision regime. Phys. Rev. E 95 (2017) 052113. https://doi.org/10.1103/PhysRevE.95.052113
A.G. Magner, M.I. Gorenstein, U.V. Grygoriev. Ultrasonic waves in classical gases. Phys. Rev. E 96, 062142 (2017). https://doi.org/10.1103/PhysRevE.96.062142
D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlations Functions (Benjamin, 1975).
D. Zubarev, V. Morozov, G. R¨opke. Statistical Mechanics of Non-Equilibrium Processes (Fizmatlit, 2002), Vol. II (in Russian).
V.P. Silin. Introduction to the Kinetic Theory of Gases (Nauka, 1971) (in Russian).
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