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Transient Propagation of Spherical Waves in Porous Material: Application of Fractional Calculus

Zine El Abiddine Fellah 1 Mohamed Fellah 2 Rémi Roncen 3 Nicholas Ongwen 4 Erick Ogam 1 Claude Depollier 5
1 O&I - Ondes et Imagerie
LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]
2 LPT - Laboratoire de Physique Théorique [University of Science and Technology Houari Boumediene]
3 ONERA / DMPE, Université de Toulouse [Toulouse]
4 Department of Physics and Materials Science, Maseno University, Maseno, Kenya.
5 LAUM - Laboratoire d'Acoustique de l'Université du Mans
Abstract : A fractional-order wave equation is established and solved for a space of three dimensions using spherical coordinates. An equivalent fluid model is used in which the acoustic wave propagates only in the fluid saturating the porous medium; this model is a special case of Biot’s theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The basic solution of the wave equation is obtained in the time domain by analytically calculating Green’s function of the porous medium and using the properties of the Laplace transforms. Fractional derivatives are used to describe, in the time domain, the fluid–structure interactions, which are of the inertial, viscous, and thermal kind. The solution to the fractional-order wave equation represents the radiation field in the porous medium emitted by a point source. An important result obtained in this study is that the solution of the fractional equation is expressed by recurrence relations that are the consequence of the modified Bessel function of the third kind, which represents a physical solution of the wave equation. This theoretical work with analytical results opens up prospects for the resolution of forward and inverse problems allowing the characterization of a porous medium using spherical waves.
Keywords : Transient Propagation of porous materials fractional calculus spherical wave viscous and thermal surfaces dynamic tortuosity and compressibility
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https://hal.archives-ouvertes.fr/hal-03543610
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Soumis le : mercredi 26 janvier 2022 - 10:18:44
Dernière modification le : jeudi 31 mars 2022 - 16:15:46
Archivage à long terme le : : mercredi 27 avril 2022 - 18:24:19

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UNIV-LEMANS | LMA_UPR7051 | UNIV-AMU | LAUM | EC-MARSEILLE | CNRS | ONERA

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Zine El Abiddine Fellah, Mohamed Fellah, Rémi Roncen, Nicholas Ongwen, Erick Ogam, et al.. Transient Propagation of Spherical Waves in Porous Material: Application of Fractional Calculus. Symmetry, MDPI, 2022, 14, ⟨10.3390/sym14020233⟩. ⟨hal-03543610⟩

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