# Acoustics of Fractal Porous Material and Fractional Calculus

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## Abstract

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## 1. Introduction

## 2. Product Measure

## 3. Acoustic Equations for Propagation in Rigid Porous Material

## 4. Fractal Porous Material and Fractional Calculus

## 5. Solution of Fractional Propagation Equation in Fractal Porous Material

## 6. Slab of Fractal Porous Material

## 7. Reflection and Transmission Coefficients

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Mandelbrot, B.B. The Fractal Geometry of Nature, updated and augmented ed.; W. H. Freeman: New York, NY, USA, 1983; 468p. [Google Scholar]
- Barnsley, M.F. Fractals Everywhere; Morgan Kaufmann: San Mateo, CA, USA, 1993. [Google Scholar]
- Williams, J.K.; Dawe, R.A. Fractals: An overview of potential applications to transport in porous media. Transp. Porous Media
**1986**, 1, 201–209. [Google Scholar] [CrossRef] - Falconer, K.F. The Geometry of Fractal Sets; Cammbridge University Press: Cammbridge, UK, 1985. [Google Scholar]
- Feder, J. Fractals; Plenum Press: New York, NY, USA, 1988. [Google Scholar]
- Balankin, A.S. Fractional space approach to studies of physical phenomena on fractals and in confined low-dimensional systems. Chaos Solitons Fractals
**2020**, 132, 109572. [Google Scholar] [CrossRef] - Tarasov, V.E. Anisotropic Fractal Media by Vector calculus in non-integer dimensional space. J. Math. Phys.
**2014**, 55, 083510. [Google Scholar] [CrossRef] [Green Version] - Tarasov, V.E. Vector calculus in non-integer dimensional space and itss applications to fractal media. Commun. Nonlinear Sci. Numer. Simul.
**2015**, 20, 360–374. [Google Scholar] [CrossRef] [Green Version] - Kugami, J. Analysis on Fractals; Cammbridge University Press: Cammbridge, UK, 2001. [Google Scholar]
- Strichartz, R.S. Differential Equations on Fractals; Princeton University Press: Princeton, NJ, USA; Oxford, UK, 2006. [Google Scholar]
- Strichartz, R.S. Analysis on Fractals. Not. Am. Math. Soc.
**1999**, 10, 1199–1208. [Google Scholar] - Harrison, J. Flux across nonsmooth boundaries and fractal Gauss/Geren/Stokes’ theorems. J. Phys. A
**1999**, 32, 5317–5328. [Google Scholar] [CrossRef] - Kumagai, T. Recent developments of analysis on fractals. In Selected Papers on Analysis and Related Topics; Amercian Mathematical Society Series 2; American Mathematical Society: Providence, RI, USA, 2008; Volume 223, pp. 81–96. [Google Scholar]
- Derfel, G.; Grabner, P.; Vogl, F. Laplace operators on fractals and related functional equations. J. Phys. A
**2012**, 45, 463001. [Google Scholar] [CrossRef] [Green Version] - Carpinteri, A.; Chiaia, B.; Cornetti, P. Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput. Methods Appl. Mech. Eng.
**2001**, 191, 3–19. [Google Scholar] [CrossRef] - Carpinteri, A.; Cornetti, P. A fractional calculus approach to the description of stress and strain localization in fractal media. Chaos Solitons Fractals
**2002**, 13, 85–94. [Google Scholar] [CrossRef] - Carpinteri, A.; Chiaia, B.; Cornetti, O. On the mechanics of quasi-brittle materials with a fractal microstructure. Eng. Fract. Mech.
**2003**, 15, 2321–2349. [Google Scholar] [CrossRef] - Tarasov, V.E. Continuum medium model for fractal media. Phys. Lett. A
**2005**, 336, 167–174. [Google Scholar] [CrossRef] [Green Version] - Tarasov, V.E. Wave equation for fractal solid string. Mod. Phys. Lett. B
**2005**, 15, 721–728. [Google Scholar] [CrossRef] [Green Version] - Tarasov, V.E. Fractional hydrodynamic equations for fractal media. Ann. Phys.
**2005**, 318, 286–307. [Google Scholar] [CrossRef] [Green Version] - Tarasov, V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particules, Fields and Media; Springer: New York, NY, USA, 2011. [Google Scholar]
- Demmie, P.N.; Ostoja-Starzewski, M. Waves in fractal media. J. Elast.
**2011**, 104, 187–204. [Google Scholar] [CrossRef] - Joumaa, H.; Ostoja-Starzewski, M. Acoustic-elastoodynamic interaction in isotropic fractal media. Eur. Phys. J. Spec. Top.
**2013**, 222, 1951–1960. [Google Scholar] [CrossRef] - Ostoja-Starzewski, M. Electromagnetism on anisotropic fractals. Zeitschrift für Angewandte Mathematik and Physik (J. Appl. Math. Mech.)
**2013**, 64, 381–390. [Google Scholar] - Ostoja-Starzewski, M.; Li, J.; Joumaa, H.; Demmie, P.N. From fractal media to continuum mechanics. Zeitschrift für Angewandte Mathematik and Physik
**2014**, 94, 373–401. [Google Scholar] [CrossRef] - Tarasov, V.E. Fractional generalisation of Liouville equations. Chaos
**2004**, 14, 123–127. [Google Scholar] [CrossRef] [Green Version] - Tarasov, V.E. Fractional Liouville and BBGKI equations. J. Phys. Conf. Ser.
**2005**, 7, 17–33. [Google Scholar] [CrossRef] [Green Version] - Calcagni, G. Quandtum filed theory, gravity and cosmology in a fractal universe. J. High Energy Phys.
**2010**, 120, 120–158. [Google Scholar] [CrossRef] [Green Version] - Calcagni, G. Geometry of fractional spaces. Adv. Theor. Math. Phys.
**2012**, 16, 549–644. [Google Scholar] [CrossRef] - Calcagni, G.; Nardelli, G. Momentum transforms and Laplacians in fractional spaces. Adv. Theor. Math. Phys.
**2012**, 16, 1315–1348. [Google Scholar] [CrossRef] [Green Version] - Calcagni, G.; Nardelli, G. Spectral dimension and diffusion in multi-scale spacetimes. Phys. Rev. D
**2013**, 88, 124025. [Google Scholar] [CrossRef] [Green Version] - Tarasov, V.E. Fractional systems and fractional Bogoliubov hierarchy equations. Phys. Rev. E
**2005**, 71, 0111102. [Google Scholar] [CrossRef] [Green Version] - Wilson, K.G. Quandtum field—Theory models in less than 4 dimensions. Phys. Rev. D
**1973**, 10, 2911–2926. [Google Scholar] [CrossRef] - Stillinger, F.H. Axiomatic basis for spaces with non-integer dimensions. J. Math. Phys.
**1977**, 18, 1224–1234. [Google Scholar] [CrossRef] - Collins, J.C. Renormalization; Cambridge University Press: Cambridge, UK, 1984. [Google Scholar]
- ’t Hooft, G.; Veltman, M. Regularization and renormalization og gauge fields. Nucl. Phys. B
**1972**, 44, 189–213. [Google Scholar] [CrossRef] [Green Version] - Leibbrandt, G. Introduction to the technique of dimensional regularization. Rev. Mod. Phys.
**1975**, 47, 849–876. [Google Scholar] [CrossRef] - Wilson, K.G.; Fisher, M.E. Critical exponents in 3.99 dimensions. Phys. Rev. Lett.
**1972**, 28, 240–243. [Google Scholar] [CrossRef] - Wilson, K.G.; Kogut, J. The renormalization group and the ϵ expansion. Phys. Rep.
**1974**, 12, 75–199. [Google Scholar] [CrossRef] - Palmer, C.; Stavrinou, P.N. Equations of motion in a non-integer-dimensional space. J. Phys. A
**2004**, 37, 6987–7003. [Google Scholar] [CrossRef] - He, X.-F. Excitons in anisotropic solids: The model of fractional-dimensional space. Phys. Rev. B
**1991**, 43, 2063–2069. [Google Scholar] [CrossRef] - De Dios-Leyva, M.; Bruno-Alfonso, A.; Matos-Abiague, A.; Oliveira, L.E. Fractional dimensional space and applications in quantum-confined semiconducting heterostructures. J. Appl. Phys.
**1997**, 82, 3155–3157. [Google Scholar] [CrossRef] - Thilagham, A. Pauli bloking effects in quantum wells. Phys. Rev. B
**1999**, 59, 3027–3032. [Google Scholar] [CrossRef] - Muslih, S.I.; Agrawal, O.P. A scaling method and its applications to problems in fractional dimensional space. J. Math. Phys.
**2009**, 50, 123501. [Google Scholar] [CrossRef] [Green Version] - Muslih, S.I.; Agrawal, O.P. Shrödinger equation in fractional space. In Fractional dynamics and Control; Baleanu, D., Tenreiro Machado, J.A., Luo, A.C.J., Eds.; Springer: New York, NY, USA, 2012; Chapter 17; pp. 209–215. [Google Scholar]
- Muslih, S.I.; Baleanu, D. Fractional multipoles in fractional space. Nonlinear Anal. Real World Appl.
**2007**, 8, 198–203. [Google Scholar] [CrossRef] - Zubair, M.; Mughal, M.J.; Naqvi, Q.A. Electromagnetic Fileds and Waves in Fractional Dimensional Space; Springer: Berlin, Germany, 2012. [Google Scholar]
- Zubair, M.; Mughal, M.J.; Naqvi, Q.A. An exact solution of the spherical wave equation in D-dimensional fractional space. J. Electromagn. Waves Appl.
**2011**, 25, 1481–1491. [Google Scholar] [CrossRef] - Lucena, L.S.; da Silva, L.R.; Tateishi, A.A.; Lenzi, M.K.; Ribeiro, H.V.; Lenzi, E.K. Solutions for a fractional diffusion equation with noninteger dimensions. Nonlinear Anal. Real World Appl.
**2012**, 13, 1955–1960. [Google Scholar] [CrossRef] - Sadallah, M.; Muslih, S.I.; Baleanu, D. Equations of motion for Einstein’s field in non-integer dimensional space. Czechoslov. J. Phys.
**2006**, 56, 323–328. [Google Scholar] [CrossRef] - Sadallah, M.; Muslih, S.I. Solution of the equations of motion for Einstein’s field in fractional D dimenion space-time. Int. J. Theor. Phys.
**2009**, 48, 3312–3318. [Google Scholar] [CrossRef] - Balankin, A.S.; Bory-Reyes, J.; Shapiro, M. Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric. Physica A
**2016**, 444, 345–359. [Google Scholar] [CrossRef] - Svozil, K. Quantum field theory on fractal spacetime: A new regularization method. J. Phys. A
**1987**, 20, 3861–3875. [Google Scholar] [CrossRef] - Fellah, Z.E.A.; Depollier, C. Transient acoustic wave propagation in rigid porous media: A time-domain approach. J. Acoust. Soc. Am.
**2000**, 107, 683–688. [Google Scholar] [CrossRef] - Allard, J.F. Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials; Chapman and Hall: London, UK, 1993. [Google Scholar]
- Biot, M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. J. Acoust. Soc. Am.
**1956**, 28, 168–178. [Google Scholar] [CrossRef] - Biot, M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. J. Acoust. Soc. Am.
**1956**, 28, 179–191. [Google Scholar] [CrossRef] - Johnson, D.L.; Koplik, J.; Dashen, R. Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J. Fluid Mech.
**1987**, 176, 379–402. [Google Scholar] [CrossRef] - Lafarge, D.; Lemarinier, P.; Allard, J.-F.; Tarnow, V. Dynamic compressibility of air in porous structures at audible frequencies. J. Acoust. Soc. Am.
**1997**, 102, 1995–2006. [Google Scholar] [CrossRef] [Green Version] - Fellah, Z.E.A.; Fellah, M.; Lauriks, W. Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material. J. Acoust. Soc. Am.
**2003**, 113, 61–72. [Google Scholar] [CrossRef] [PubMed] - Balankin, A.S.; Valdivia, J.-C.; Marquez, J.; Susarrey, O.; Solorio-Avila, M.A. Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity. Phys. Lett. A
**2016**, 380, 2767–2773. [Google Scholar] [CrossRef] - Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach Science Publishers: Amsterdam, The Netherlands, 1993; pp. 93–112. [Google Scholar]
- Fellah, Z.E.A.; Fellah, M.; Lauriks, W.; Depollier, C.; Chapelon, J.Y.; Angel, Y.C. Solution in time domain of ultrasonic propagation equation in a porous material. Wave Motion
**2003**, 38, 151–163. [Google Scholar] [CrossRef] - Tarasov, V.E. Acoustic waves in fractal media: Non-integer dimensional spaces approach. Wave Motion
**2016**, 63, 18–22. [Google Scholar] [CrossRef] - Fellah, M.; Fellah, Z.E.A.; Berbiche, A.; Ogam, E.; Mitri, F.G.; Depollier, D. Transient ultrasonic wave propagation in porous material of non-integer space dimension. Wave Motion
**2017**, 72, 276–286. [Google Scholar] [CrossRef]

**Figure 2.**Variation of the equivalent thickness (${L}^{\alpha}{\pi}^{\alpha /2}/\alpha \Gamma (\alpha /2$) in the fractal material as a function of the fractal dimension.

**Figure 3.**Variation of the front wave velocity in the fractal material as a function of the fractal dimension.

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**MDPI and ACS Style**

Fellah, Z.E.A.; Fellah, M.; Ongwen, N.O.; Ogam, E.; Depollier, C.
Acoustics of Fractal Porous Material and Fractional Calculus. *Mathematics* **2021**, *9*, 1774.
https://doi.org/10.3390/math9151774

**AMA Style**

Fellah ZEA, Fellah M, Ongwen NO, Ogam E, Depollier C.
Acoustics of Fractal Porous Material and Fractional Calculus. *Mathematics*. 2021; 9(15):1774.
https://doi.org/10.3390/math9151774

**Chicago/Turabian Style**

Fellah, Zine El Abiddine, Mohamed Fellah, Nicholas O. Ongwen, Erick Ogam, and Claude Depollier.
2021. "Acoustics of Fractal Porous Material and Fractional Calculus" *Mathematics* 9, no. 15: 1774.
https://doi.org/10.3390/math9151774