On the dispersion relation for compressible Navier-Stokes Equations

TL;DR

This paper revisits classical theories of sound dispersion and attenuation in non-ideal fluids, reformulating conditions for propagation at the isothermal speed and providing asymptotic analysis for different physical regimes.

Contribution

It offers a reformulation of Fletcher's analysis, clarifies conditions for isothermal sound propagation, and develops asymptotic formulas for dispersion and attenuation in compressible Navier-Stokes flows.

Findings

Conditions for sound to propagate at the isothermal speed of sound.

Asymptotic formulas for dispersion and attenuation.

Clarification of physical regimes for different formulas.

Abstract

In this paper we revisit the classical sound dispersion and attenuation theory due to Stokes [5], 1845, and Kirchhoff [3], 1868, for the propagation of sound in non-ideal fluids. In particular we reformulate the analysis due to Fletcher[2], 1974, showing conditions for which the sound propagates at the isothermal speed of sound. Also we presents asymptotic developments making precise the physical conditions under which the different dispersion and attenuation formulas apply.The more complex case of two-fluid flow is addressed by Benjelloun and Ghidaglia [1] to which the reader is referred.