Influence of viscosity and thermal conductivity on propagation of sound impulses

Abstract

The first investigation of sound wave attenuation in a viscous gas is due to Stokes, the effect of thermal conductivity was taken into account later by Kirchhoff. A quite complete presentation of their results is available in the classic monograph of Lord Rayleigh [1], From the very beginning Stokes, Just as Kirchhoff, used the linear equations of acoustics as the starting point, and not the exact equations which describe the motion of continuous media.In his studies of propagation of olane sound impulses in an ideal gas (without viscosity and thermal conductivity) Crussard showed [2] that asymptotic relationships of shock wave decay at large distances from the location of their origin are different from those for accoustic waves. Correct derivation of these relationships is not possible without consideration of non-linear terms in equations of gas dynamics. Extension of Crussard's theory to cylindrical and spherical shock waves was given by Landau [3], Khristianovich [4], Sedov [5] and Whitham [6] using different methods.It follows from the paper of Taylor [7] that the structure of weak shock waves is determined basically by corrective processes, related to the non-linear nature of Navier-Stokes equations, and by dissipation of energy at the expense of viscosity and thermal conductivity of real media. Therefore it appeared natural that both factors mentioned will influence the propagation of sound impulses to the same extent. This point of view was expressed by Lighthill [8]. He made a detailed analysis of this concept using plane motion as his example.The decay of perturbations in cylindrical and spherical sound impulses is examined below. It turns out that the structure of waves and asymptotic relationships of their decay when time t → ∞ are related to effects of viscosity and thermal conductivity. At this stage of the process, consideration of nonlinear terms in the Navier-Stokes equations may be neglected because their influence on the formation of the flow field is negligibly small. Variation of all gas parameters within the impulses occurs smoothly, shock waves are absent. Conversely, the motion of shock waves, as long as their width is much smaller than the general length of the wave, is determined by nonlinear convective terms of equations of gas dynamics.When t → ∞ the change of the maximum value of the excess pressure in N-waves, with consideration of viscosity and thermal conductivity, is inversely proportional to ∼ for motions with axial symmetry and to t2 for cenrally symmetric motions. The assertion of Lighthill [8] that asymptotic relationships of decay of perturbations must be exponential, turned out to be incorrect; the excess pressure varies according to an exponential law only in periodic sound waves with fixed wave length [1]. The conclusions obtained are based on a generalization of analysis of short waves carried out by Khristianovich [4] for unsteady one-dimensional motion of an ideal gas.