Evolution of Perturbation in Quiescent Medium

TL;DR

This paper derives a perturbation equation for a dissipative medium from first principles, revealing a wave number-dependent transition from dispersive to diffusive behavior, which has not been previously reported.

Contribution

It introduces a novel theoretical framework for understanding perturbation evolution in dissipative media without relying on Stokes's hypothesis.

Findings

Identification of a cut-off wave number for dissipative behavior

Derivation of dispersion relations for 1D and 3D perturbations

Discovery of a transition from dispersive to diffusive dynamics

Abstract

Here, the perturbation equation for a dissipative medium is derived from the first principle from the linearized compressible Navier-Stokes equation without Stokes's hypothesis. The dispersion relations of this generic governing equation are obtained for one and three-dimensional perturbations, which exhibit both the dispersive and dissipative nature of the perturbations traveling in a dissipative medium, depending upon the length scale. We specifically provide a theoretical cut-off wave number above which the perturbation equation represents diffusive and dissipative nature. Such behavior has not been reported before, as per the knowledge of the authors.