A Fractional Acoustic Wave Equation from Multiple Relaxation Loss and Conservation Laws

TL;DR

This paper derives a fractional acoustic wave equation from multiple relaxation mechanisms and conservation laws, establishing a connection with the fractional Zener model to better understand frequency power-law attenuation in acoustics.

Contribution

It introduces a link between the fractional Zener wave equation and the multiple relaxation framework, unifying two approaches to modeling acoustic attenuation.

Findings

The fractional Zener wave equation can be derived from a continuous distribution of relaxation mechanisms.

The two models are equivalent under certain conditions, confirming their physical consistency.

The work enhances understanding of frequency power-law attenuation in acoustical wave propagation.

Abstract

This work concerns causal acoustical wave equations which imply frequency power-law attenuation. A connection between the five-parameter fractional Zener wave equation, which is derived from a fractional stress-strain relation plus conservations of mass and momentum, and the physically well established multiple relaxation framework is developed. It is shown that for a certain continuous distribution of relaxation mechanisms, the two descriptions are equal.