Linking multiple relaxation, power-law attenuation, and fractional wave equations

2011/11/01 by Sven Peter Näsholm, Sverre Holm · 5 citations
Engineering · Mathematics · #Acoustic Wave Phenomena Research #Fractional Differential Equations Solutions #Acoustic Wave Resonator Technologies #Attenuation #Fractional calculus #Wave equation #Relaxation (psychology) #Power law #Acoustic attenuation #Physics #Compressibility #Acoustic wave equation #Acoustic wave #Acoustics #Mathematical analysis #Mathematics #Mechanics #Optics

paper · doi:10.1121/1.3641457

openalex publication_date 2011/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25

Abstract

The acoustic wave attenuation is described by an experimentally established frequency power law in a variety of complex media, e.g., biological tissue, polymers, rocks, and rubber. Recent papers present a variety of acoustical fractional derivative wave equations that have the ability to model power-law attenuation. On the other hand, a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by Nachman et al. [J. Acoust. Soc. Am. 88, 1584-1595 (1990)]. Through assumption of a continuum of relaxation mechanisms, each with an effective compressibility described by a distribution related to the Mittag-Leffler function, this paper shows that the wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative wave equation. This work therefore provides a physically based motivation for use of fractional wave equations in acoustic modeling.

Citations

• Über den Fundamentalsatz in der Teorie der Funktionen Ea(x)
• Linear Models of Dissipation whose Q is almost Frequency Independent--II
• Fractional calculus - A different approach to the analysis of viscoelastically damped structures
• Fractional Differential Equations

Cited by

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• Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations
• New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model
• On a fractional Zener elastic wave equation
• Fractional calculus [wikipedia]

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