Acoustic Equation in a Lossy Medium

TL;DR

This paper derives an acoustic equation for lossy media from fundamental principles, revealing a novel dispersive and diffusive behavior dependent on length scale, including a theoretical cut-off wave number for diffusion.

Contribution

It introduces a new derivation of the acoustic equation in lossy media without Stokes' hypothesis, highlighting a previously unreported diffusive behavior at certain scales.

Findings

Dispersion relation shows dispersive and dissipative nature of acoustic waves.

Identification of a theoretical cut-off wave number for diffusive behavior.

Novel insight into acoustic wave behavior in lossy media.

Abstract

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