mathematics Article Acoustics of Fractal Porous Material and Fractional Calculus Zine El Abiddine Fellah 1,* , Mohamed Fellah 2, Nicholas O. Ongwen 3 , Erick Ogam 1 and Claude Depollier 4,5 1 LMA, CNRS, UMR 7031, Centrale Marseille, Aix-Marseille Univ, CEDEX 20, 13402 Marseille, France;
[email protected] 2 Laboratoire de Physique Théorique, Faculté de Physique, USTHB, BP 32 El Alia, Bab Ezzouar 16111, Algeria;
[email protected] 3 Department of Physics and Materials Science, Maseno University, Maseno 40105, Kenya;
[email protected] 4 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;
[email protected] 5 ESST, 43 Chemin Sidi M’Barek, Oued Romane, El Achour 16104, Algeria * Correspondence:
[email protected] 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 Citation: Fellah, Z.E.A.; Fellah, M.; of the material.