Fractional Calculus and Applied Analysis

, Volume 16, Issue 1, pp 26–50 | Cite as

On a fractional Zener elastic wave equation

Survey Paper
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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.

Key Words and Phrases

fractional calculus acoustical wave equations elastic wave equations fractional wave equations fractional viscoelasticity fractional ordinary and partial differential equations 

MSC 2010

Primary 26A33 Secondary 33E12, 34A08, 34K37, 35L05, 92C50, 92C55, 35R11, 74J10 

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© Versita Warsaw and Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of OsloOsloNorway

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