{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T07:23:06Z","timestamp":1775373786420,"version":"3.50.1"},"reference-count":67,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:00:00Z","timestamp":1663718400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/sym14101971","type":"journal-article","created":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T04:07:07Z","timestamp":1663906027000},"page":"1971","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus"],"prefix":"10.3390","volume":"14","author":[{"given":"Djihane","family":"Benmorsli","sequence":"first","affiliation":[{"name":"Laboratory of Theoretical Physics, Faculty of Physics, University of Science and Technology Houari-Boumediene, BP 32 El Alia, Bab Ezzouar 16111, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9682-0876","authenticated-orcid":false,"given":"Zine El Abiddine","family":"Fellah","sequence":"additional","affiliation":[{"name":"Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Laboratory of Mechanics and Acoustics, CEDEX 20, 13402 Marseille, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Djema","family":"Belgroune","sequence":"additional","affiliation":[{"name":"Materials Physics Laboratory, Faculty of Physics, University of Science and Technology Houari-Boumediene, BP 32 El Alia, Bab Ezzouar 16111, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicholas O.","family":"Ongwen","sequence":"additional","affiliation":[{"name":"Department of Physics and Materials Science, Maseno University, Maseno 40105, Kenya"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7012-6504","authenticated-orcid":false,"given":"Erick","family":"Ogam","sequence":"additional","affiliation":[{"name":"Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Laboratory of Mechanics and Acoustics, CEDEX 20, 13402 Marseille, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5024-7066","authenticated-orcid":false,"given":"Claude","family":"Depollier","sequence":"additional","affiliation":[{"name":"Laboratoire d\u2019Acoustique de l\u2019Universit\u00e9 du Mans (LAUM), UMR 6613, Institut d\u2019Acoustique, CNRS, Le Mans Universit\u00e9, Avenue O. Messiaen, CEDEX 09, 72085 Le Mans, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohamed","family":"Fellah","sequence":"additional","affiliation":[{"name":"Laboratory of Theoretical Physics, Faculty of Physics, University of Science and Technology Houari-Boumediene, BP 32 El Alia, Bab Ezzouar 16111, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"S11","DOI":"10.1016\/S0020-1383(16)47003-8","article-title":"Bone mechanical properties and changes with osteoporosis","volume":"47","author":"Osterhoff","year":"2016","journal-title":"Injury"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/0140-6736(93)92555-8","article-title":"Bone density at various sites for prediction of hip fractures","volume":"341","author":"Cummings","year":"1993","journal-title":"Lancet"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"355","DOI":"10.7326\/0003-4819-111-5-355","article-title":"Baseline Measurement of Bone Mass Predicts Fracture in White Women","volume":"111","author":"Hui","year":"1989","journal-title":"Ann. 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