On a Fractional Zener Elastic Wave Equation

TL;DR

This paper reviews a fractional Zener elastic wave equation that models frequency-dependent attenuation in elastic waves, connecting fractional calculus, viscoelastic models, and acoustical applications including medical elastography.

Contribution

It synthesizes recent results on fractional calculus modeling in acoustics and elastography, emphasizing the physical basis and applications in medical fields.

Findings

The wave equation exhibits three distinct power-law attenuation regimes.

It links fractional stress-strain relations with classical relaxation models.

The model applies to medical elastography and acoustics.

Abstract

This survey concerns a causal elastic wave equation which implies frequency power-law attenuation. The wave equation can be derived from a fractional Zener stress-strain relation plus linearized conservation of mass and momentum. A connection between this four-parameter fractional wave equation and a physically well established multiple relaxation acoustical wave equation is reviewed. The fractional Zener wave equation implies three distinct attenuation power-law regimes and a continuous distribution of compressibility contributions which also has power-law regimes. Furthermore it is underlined that these wave equation considerations are tightly connected to the representation of the fractional Zener stress-strain relation, which includes the spring-pot viscoelastic element, and by a Maxwell-Wiechert model of conventional springs and dashpots. A purpose of the paper is to make available recently published results on fractional calculus modeling in the field of acoustics and elastography, with special focus on medical applications.