Acoustics of Fractal Porous Material and Fractional Calculus

Zine El Abiddine Fellah, Mohamed Fellah, Nicholas O. Ongwen, Erick Ogam and Claude Depollier
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Zine El Abiddine Fellah: LMA, CNRS, UMR 7031, Centrale Marseille, Aix-Marseille Univ, CEDEX 20, 13402 Marseille, France
Mohamed Fellah: Laboratoire de Physique Théorique, Faculté de Physique, USTHB, BP 32 El Alia, Bab Ezzouar 16111, Algeria
Nicholas O. Ongwen: Department of Physics and Materials Science, Maseno University, Maseno 40105, Kenya
Erick Ogam: LMA, CNRS, UMR 7031, Centrale Marseille, Aix-Marseille Univ, CEDEX 20, 13402 Marseille, France
Claude Depollier: UMR CNRS 6613, Laboratoire d’Acoustique de l’Universite du Maine, LUNAM Universite du Maine, UFR STS Avenue O. Messiaen, CEDEX 09, 72085 Le Mans, France

Mathematics, 2021, vol. 9, issue 15, 1-16

Abstract: In this paper, we present a fractal (self-similar) model of acoustic propagation in a porous material with a rigid structure. The fractal medium is modeled as a continuous medium of non-integer spatial dimension. The basic equations of acoustics in a fractal porous material are written. In this model, the fluid space is considered as fractal while the solid matrix is non-fractal. The fluid–structure interactions are described by fractional operators in the time domain. The resulting propagation equation contains fractional derivative terms and space-dependent coefficients. The fractional wave equation is solved analytically in the time domain, and the reflection and transmission operators are calculated for a slab of fractal porous material. Expressions for the responses of the fractal porous medium (reflection and transmission) to an acoustic excitation show that it is possible to deduce these responses from those obtained for a non-fractal porous medium, only by replacing the thickness of the non-fractal material by an effective thickness depending on the fractal dimension of the material. This result shows us that, thanks to the fractal dimension, we can increase (sometimes by a ratio of 50) and decrease the equivalent thickness of the fractal material. The wavefront speed of the fractal porous material depends on the fractal dimension and admits several supersonic values. These results open a scientific challenge for the creation of new acoustic fractal materials, such as metamaterials with very specific acoustic properties.

Keywords: fractal media; porous material; fractional calculus; wave equation; acoustic wave; reflection and transmission (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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