On the sound dispersion and attenuation in fluids due to thermal and viscous effects

TL;DR

This paper derives a second-order dispersion relation for sound waves in viscous, heat-conducting fluids, clarifying classical attenuation formulas and addressing recent controversies in the literature.

Contribution

It provides a complete derivation of the classical Stokes-Kirchhoff attenuation formula from the Navier-Stokes system and clarifies misconceptions in recent alternative derivations.

Findings

Derived a second-order dispersion relation for sound in viscous fluids.

Reestablished the classical Stokes-Kirchhoff attenuation formula.

Clarified differences in dispersion and attenuation formulas in literature.

Abstract

In this paper, we derive a dispersion relation for sound waves in viscous and heat conducting fluids. In particular this dispersion (i.e. variation of speed of sound with frequency) is shown to be of second order of magnitude, w.r.t. Knudsen numbers, as in the Stokes [2] case, corresponding to non-conductive fluid (Prandtl number P r = $\infty$). This formula completes the classical attenuation relation called Stokes-Kirchhoff. We represent in a simplified manner the Kirchhoff approach to derive this attenuation [1], starting from the 3D compressible Navier-stokes system. The classical Stokes-Kirchhoff formula has been questioned recently in [3] and a different (and incorrect) formula was proposed. We point out the non-trivial assumptions that are violated in the new derivation in [3] to reestablish the classical Stokes-Kirchhoff formula. Finally, we give an explanation to differences in dispersion and attenuation formulae that one may find in the literature through analysing the form of the considered attenuated solutions.