Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus
Djihane Benmorsli,
Zine El Abiddine Fellah,
Djema Belgroune,
Nicholas O. Ongwen,
Erick Ogam,
Claude Depollier,
Mohamed Fellah
Affiliations
Djihane Benmorsli
Laboratory of Theoretical Physics, Faculty of Physics, University of Science and Technology Houari-Boumediene, BP 32 El Alia, Bab Ezzouar 16111, Algeria
Zine El Abiddine Fellah
Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Laboratory of Mechanics and Acoustics, CEDEX 20, 13402 Marseille, France
Djema Belgroune
Materials Physics Laboratory, Faculty of Physics, University of Science and Technology Houari-Boumediene, BP 32 El Alia, Bab Ezzouar 16111, Algeria
Nicholas O. Ongwen
Department of Physics and Materials Science, Maseno University, Maseno 40105, Kenya
Erick Ogam
Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Laboratory of Mechanics and Acoustics, CEDEX 20, 13402 Marseille, France
Claude Depollier
Laboratoire d’Acoustique de l’Université du Mans (LAUM), UMR 6613, Institut d’Acoustique, CNRS, Le Mans Université, Avenue O. Messiaen, CEDEX 09, 72085 Le Mans, France
Mohamed Fellah
Laboratory of Theoretical Physics, Faculty of Physics, University of Science and Technology Houari-Boumediene, BP 32 El Alia, Bab Ezzouar 16111, Algeria
In this paper, the influence of the transverse wave on sound propagation in a porous medium with a flexible structure is considered. The study is carried out in the time domain using the modified Biot theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The viscous exchanges between the fluid and the structure are described by fractional calculus. When a sound pulse arrives at normal incidence on a porous material with a flexible structure, the transverse waves interfere with the longitudinal waves during propagation because of the viscous interactions that appear between the fluid and the structure. By performing a calculation in the Laplace domain, the reflection and transmission operators are derived. Their time domain expressions depend on the Green functions of the longitudinal and transverse waves. In order to study the effects of the transverse wave on the transmitted longitudinal waves, numerical simulations of the transmitted waves in the time domain by varying the characteristic parameters of the medium are realized whether the transverse wave is considered or not.
