The propagation of sound pulses from a point source in a medium where the index of refraction varies randomly is studied by means of the Born approximation to the wave equation. The coefficient of variation (standard deviation of the amplitude of a series of pulses, expressed as a percentage of the mean amplitude of the series) is evaluated for pulse lengths short compared with the time in which the refractive index varies significantly, and for ranges large compared with the wavelength of the sound. The results are in agreement with the experiments of Sheehy.
A three-fluid theory (using Maxwell's equations together with a set of coupled hydrodynamic equations for an interacting mixture of electrons, ions, and neutral molecules) is used to examine small-amplitude oscillations in an infinite, homogeneous, partly ionized gas with a uniform external magnetic field. Dispersion relations are obtained when the uniform magnetic field is either parallel or perpendicular to the direction of wave propagation, and the results are interpreted (in the case of negligible collisional damping) by analyzing plots of phase velocity versus frequency. Comparison is made with the simpler one-fluid theories (the magneto-ionic theory, magnetohydrodynamics, and acoustics), as well as with the two-fluid theory for a fully ionized gas. Collisional damping effects are included in all the dispersion relations and are described in detail for the magneto-ionic and magnetohydrodynamic limits.
Solutions of the Boltzmann equation of the form f(r, v, t) = f(0)(r, v, t) Σ ap(r, t)Mp(v; r, t) are considered, where f(0) is some zero-order approximation to f. It is shown that, to obtain reasonable equations for the ap and to relate them to the moments of f, the Mp should be a polynomial set in velocity-space, orthogonalized with f(0) as the weighting factor. It is shown how the Burnett method and the Grad method are derivable when the problem is a near-equilibrium one; and how the half-range polynomial method results when particular symmetry conditions are imposed. An illustration of the method is given for the case of the equilibration of a two-temperature gas mixture.
The problem of sound propagation in highly rarefied monatomic gases is investigated from the point of view of general orthogonal polynomial solutions (in velocity space) of the Boltzmann equation. It is shown that the usual expansion solutions of the Boltzmann equation (Chapman-Enskog-Burnett, and Grad) are not valid for this problem. Solutions, instead, are obtained by means of an expansion of the distribution function in a set of velocity polynomials which have been orthogonalized with respect to a zero-order distribution function characteristic of a collisionless gas, rather than a Maxwell distribution function as is usually done. These solutions are used to derive absorption and dispersion functions characterizing the sound propagation. These are shown to be in agreement with experimental data for the problem.
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