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Signal, Image and Video Processing

, Volume 6, Issue 3, pp 389–392 | Cite as

Fractional calculus approach to the acoustic wave propagation with space-dependent sound speed

  • G. Casasanta
  • R. GarraEmail author
Original Paper

Abstract

In this note we study the space-fractional wave equation in relation to the propagation of acoustic waves with space-dependent sound speed. We take into account this variability, by using the space-fractional derivative in the classical wave equation. In order to give a clear comprehension of this mathematical formulation, we discuss the analytic solution of a simple boundary value problem (BVP) by an operational method, finding a fractional oscillating spatial profile.

Keywords

Fractional differential equations Wave propagation Space-fractional wave equation Sound propagation 

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References

  1. 1.
    Chen T., Millero F.J.: Speed of sound in seawater at high pressures. J. Acoust. Soc. Am. 62(5), 1129–1135 (1977)CrossRefGoogle Scholar
  2. 2.
    Chiocci F.L., Ridente D.: Regional-scale seafloor mapping and geohazard assessment. Experience from the Italian project MaGIC (Marine Geohazards along the Italian Coasts). Mar. Geophys. Res. 32(1–2), 13–23 (2011)CrossRefGoogle Scholar
  3. 3.
    Fellah Z.E.A., Depollier C., Fellah M.: Application of fractional calculus to the sound waves propagation in rigid porous materials: validation via ultrasonic measurements. Acta Acust. United Acust. 88(1), 34–39 (2002)Google Scholar
  4. 4.
    Fellah M., Fellah Z.E.A, Depollier C.: Transient wave propagation in inhomogeneous porous materials: application of fractional derivatives. Signal Process. 86, 2658–2667 (2006)zbMATHCrossRefGoogle Scholar
  5. 5.
    Fellah M., Fellah Z.E.A, Depollier C.: Generalized hyperbolic fractional equation for transient-wave propagation in layered rigid-frame porous materials. Phys. Rev. E 77, 016601–016605 (2008)CrossRefGoogle Scholar
  6. 6.
    Garra, R., Polito, F.: Analytic solutions of fractional differential equations by operational methods. Appl. Math. Comput. (2011, to appear)Google Scholar
  7. 7.
    Mainardi F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6), 23–28 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Metzler R., Nonnenmacher T.F.: Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation. Chem. Phys. 284(1–2), 67–90 (2002)CrossRefGoogle Scholar
  9. 9.
    Podlubny I.: Fractional Differential Equations. Academic Press, New York (1999)zbMATHGoogle Scholar
  10. 10.
    Sabatier J., Agrawal O.P., Machado J.A.T.: Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)zbMATHGoogle Scholar
  11. 11.
    Tarasov V.E.: Fractional hydrodynamic equations for fractal media. Ann. Phys. 318(2), 286–307 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Dipartimento di Fisica“Sapienza” Università di RomaRomeItaly

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