Sound dispersion in single-component systems

The present paper considers the theoretical description of the propagation of sound waves in a one component monatomic gas. The interatomic potential is assumed to vary as the inverse fourth power of the interatomic separation, that is for so-called Maxwell molecules. The eigenvalues and eigenfunctions of the linearized Boltzmann collision operator are known for this model. We emphasize the behaviour of this system in the rarefied, large Knudsen number regime for which the convergence of solutions of the Boltzmann equation can be very slow. We carry out a detailed comparison of the previous formalisms by Wang Chang and Uhlenbeck [C.S. Wang Chang, G.E. Uhlenbeck, The kinetic theory of gases, in: G.E. Uhlenbeck, De Boer, (Eds.), Studies in Statistical Mechanics, vol. 5, Elsevier, New York, 1970, pp. 43–75], Alexeev [B.V. Alexeev, Philos. Trans. R. Soc. A 349 (1994) 357] and Sirovich and Thurber [L. Sirovich, J. K. Thurber, J. Math. Phys. 10 (1969) 239]. The latter exploit a general method of solution of the Boltzmann equation developed by Gross and Jackson. We demonstrate that the Generalized Boltzmann Equation proposed by Alexeev is not appropriate and we show the reasoning for the success of the Sirovich Thurber approach over the Wang Chang and Uhlenbeck calculations. Comparisons are made with experimental data.