{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T13:45:35Z","timestamp":1762868735617,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,25]],"date-time":"2022-01-25T00:00:00Z","timestamp":1643068800000},"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>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\u2019s 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\u2019s 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\u2013structure 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.<\/jats:p>","DOI":"10.3390\/sym14020233","type":"journal-article","created":{"date-parts":[[2022,1,25]],"date-time":"2022-01-25T21:07:11Z","timestamp":1643144831000},"page":"233","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Transient Propagation of Spherical Waves in Porous Material: Application of Fractional Calculus"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9682-0876","authenticated-orcid":false,"given":"Zine El Abiddine","family":"Fellah","sequence":"first","affiliation":[{"name":"Laboratory of Mechanics and Acoustics, French National Centre for Scientific Research LMA, CNRS, UMR 7031, Centrale Marseille, Aix-Marseille University CEDEX 20, F-13402 Marseille, France"}]},{"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"}]},{"given":"R\u00e9mi","family":"Roncen","sequence":"additional","affiliation":[{"name":"French Aerospace Lab, ONERA\/Multi-Physics Department for Energy, Toulouse University, F-31055 Toulouse, France"}]},{"given":"Nicholas O.","family":"Ongwen","sequence":"additional","affiliation":[{"name":"Department of Physics and Materials Science, Maseno University, Maseno 40105, Kenya"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7012-6504","authenticated-orcid":false,"given":"Erick","family":"Ogam","sequence":"additional","affiliation":[{"name":"Laboratory of Mechanics and Acoustics, French National Centre for Scientific Research LMA, CNRS, UMR 7031, Centrale Marseille, Aix-Marseille University CEDEX 20, F-13402 Marseille, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5024-7066","authenticated-orcid":false,"given":"Claude","family":"Depollier","sequence":"additional","affiliation":[{"name":"Acoustics Laboratory of the University of Le Mans, French National Centre for Scientific Research, CNRS UMR 6613, UFR STS Avenue O. Messiaen, CEDEX 09, F-72085 Le Mans, France"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,25]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Allard, J.F. (1993). Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials, Chapman and Hall.","key":"ref_1","DOI":"10.1007\/978-94-011-1866-8"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"114902","DOI":"10.1063\/1.2804127","article-title":"Measuring permeability of porous materials at low frequency range via acoustic transmitted waves","volume":"78","author":"Fellah","year":"2007","journal-title":"Rev. Sci. Instrum."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.paerosci.2014.12.003","article-title":"A review of acoustic dampers applied to combustion chambers in aerospace industry","volume":"74","author":"Zhao","year":"2015","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1074","DOI":"10.1016\/j.jsv.2010.09.032","article-title":"Non-ambiguous recovery of Biot poroelastic parameters of cellular panels using ultrasonicwaves","volume":"330","author":"Ogam","year":"2011","journal-title":"J. Sound Vib."},{"key":"ref_5","first-page":"96","article-title":"Acoustic Behavior in Three Types of Housing: Brick Social Housing, Structural Insulated Panel (SIP) Emergency Housing and Mediagua Emergency Housing","volume":"18","author":"Garay","year":"2019","journal-title":"Environ. Sci. Rev. Constr."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1017\/S0022112087000727","article-title":"Theory of dynamic permeability and tortuosity in fluid-saturated porous media","volume":"176","author":"Johnson","year":"1987","journal-title":"J. Fluid Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1995","DOI":"10.1121\/1.419690","article-title":"Dynamic compressibility of air in porous structures at audible frequencies","volume":"102","author":"Lafarge","year":"1997","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_8","first-page":"122","article-title":"Evaluation of the viscous characteristic length of air-saturated porous materials from the ultrasonic dispersion curve","volume":"322","author":"Brown","year":"1996","journal-title":"Comptes R. Acad. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2009","DOI":"10.1063\/1.363817","article-title":"Determination of the viscous and thermal characteristic lengths of plastic foams by ultrasonic measurements in helium and air","volume":"80","author":"Leclaire","year":"1996","journal-title":"J. Appl. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1121\/1.5044423","article-title":"Bayesian inference for the ultrasonic characterization of rigid porous materials using reflected waves by the first interface","volume":"144","author":"Roncen","year":"2018","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.wavemoti.2004.06.004","article-title":"Influence of dynamic tortuosity and compressibility on the propagation of transient waves in porous media","volume":"41","author":"Fellah","year":"2005","journal-title":"Wave Motion"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/S0165-2125(03)00045-3","article-title":"Solution in time domain of ultrasonic propagation equation in a porous material","volume":"38","author":"Fellah","year":"2003","journal-title":"Wave Motion"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"557","DOI":"10.3813\/AAA.918635","article-title":"Transients in porous media: Exact and modelled time-domain Green\u2019s functions","volume":"99","author":"Kergomard","year":"2013","journal-title":"Acta Acust. Acust."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1629","DOI":"10.1121\/1.5095403","article-title":"Inverse identification of a higher order viscous parameter of rigid porous media in the high frequency domain","volume":"145","author":"Roncen","year":"2019","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"084906","DOI":"10.1063\/1.2798930","article-title":"Transient acoustic wave propagation in air-saturated porous media at low frequencies","volume":"102","author":"Fellah","year":"2019","journal-title":"J. Appl. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1926","DOI":"10.1121\/1.2179749","article-title":"Measuring flow resistivity of porous materials at low frequencies range via acoustic transmitted waves","volume":"119","author":"Fellah","year":"2006","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2627","DOI":"10.1121\/1.3641402","article-title":"Measuring static thermal permeability and inertial factor of rigid porous materials","volume":"130","author":"Sadouki","year":"2011","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3084","DOI":"10.1121\/1.5080561","article-title":"Acoustical modeling and Bayesian inference for rigid porous media in the low-mid frequency regime","volume":"144","author":"Roncen","year":"2018","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1121\/1.1528592","article-title":"Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material","volume":"113","author":"Fellah","year":"2003","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"108186","DOI":"10.1016\/j.ymssp.2021.108186","article-title":"Deterministic and statistical methods for the characterisation of poroelastic media from multi-observation sound absorption measurements","volume":"163","author":"Cuenca","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"204902","DOI":"10.1063\/1.4833546","article-title":"Simultaneous determination of porosity, tortuosity, viscous and thermal characteristic lengths of rigid porous materials","volume":"114","author":"Fellah","year":"2013","journal-title":"J. Appl. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1998","DOI":"10.1121\/10.0002162","article-title":"Estimation of all six parameters of Johnson-Champoux-Allard-Lafarge model for acoustical porous materials from impedance tube measurements","volume":"148","author":"Jaouen","year":"2020","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2871","DOI":"10.1063\/1.1569412","article-title":"Inverse problem in air-saturated porous media via reflected waves","volume":"74","author":"Fellah","year":"2003","journal-title":"Rev. Sci. Instrum."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3232","DOI":"10.1121\/1.4919806","article-title":"Normalized inverse characterization of sound absorbing rigid porous media","volume":"137","author":"Zielinski","year":"2015","journal-title":"J. Acoust. Soc. Am."},{"doi-asserted-by":"crossref","unstructured":"Alruwaili, A.D., Seadawy, A.R., Ali, A., and Beinane, S.A.O. (2021). Novel Analytical Approach for the Space-Time Fractional (2+1)-Dimensional Breaking Soliton Equation via Mathematical Methods. Mathematics, 9.","key":"ref_25","DOI":"10.3390\/math9243253"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s11075-017-0306-0","article-title":"A new eight-order symmetric two-step multiderivative method for the numerical solution of second-order IVPs with oscillating solutions","volume":"77","author":"Shokri","year":"2018","journal-title":"Numer. Algorithm"},{"doi-asserted-by":"crossref","unstructured":"Bockstal, K.V. (2020). Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order). Mathematics, 8.","key":"ref_27","DOI":"10.3390\/math8081283"},{"doi-asserted-by":"crossref","unstructured":"De Rosa, S., Polimeni, A., De Velli, G., Conte, M., Sorrentino, S., Spaccarotella, C., Mongiardo, A., Sabatino, J., Contarini, M., and Todaro, D. (2019). Reliability of Instantaneous Wave-Free Ratio (iFR) for the Evaluation of Left Main Coronary Artery Lesions. J. Clin. Med., 8.","key":"ref_28","DOI":"10.3390\/jcm8081143"},{"doi-asserted-by":"crossref","unstructured":"Skudrzyk, E. (1971). Solution of the Wave Equation in General Spherical Coordinates. The Foundations of Acoustics, Springer.","key":"ref_29","DOI":"10.1007\/978-3-7091-8255-0_20"},{"doi-asserted-by":"crossref","unstructured":"Keller, J. (1964). Stochastic equations and wave propagation in random media. Stochastic Processess in Mathematical Physics and Engineering, Richard Ernest Bellman, American Mathematical Society.","key":"ref_30","DOI":"10.1090\/psapm\/016\/0178638"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1121\/1.1908239","article-title":"Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range","volume":"28","author":"Biot","year":"1956","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1121\/1.1908241","article-title":"Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range","volume":"28","author":"Biot","year":"1956","journal-title":"J. Acoust. Soc. Am."},{"unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers.","key":"ref_33"},{"unstructured":"Jeffrey, A., and Zwillinger, D. (2007). Table of Integrals, Series, and Products Academic Press, Elsevier Academic Press.","key":"ref_34"},{"unstructured":"Abramowitz, M., and Stegun, I.A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.","key":"ref_35"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/233\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:07:25Z","timestamp":1760134045000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/233"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,25]]},"references-count":35,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["sym14020233"],"URL":"https:\/\/doi.org\/10.3390\/sym14020233","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,25]]}}}