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Research Papers

Attenuated Fractional Wave Equations With Anisotropy

[+] Author and Article Information
Mark M. Meerschaert

Department of Statistics and Probability,
Michigan State University,
East Lansing, MI 48824
e-mail: mcubed@stt.msu.edu

Robert J. McGough

Department of Electrical
and Computer Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: mcgough@egr.msu.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 18, 2013; final manuscript received October 24, 2013; published online July 25, 2014. Assoc. Editor: Thomas J. Royston.

J. Vib. Acoust 136(5), 050902 (Jul 25, 2014) (5 pages) Paper No: VIB-13-1115; doi: 10.1115/1.4025940 History: Received April 18, 2013; Revised October 24, 2013

This paper develops new fractional calculus models for wave propagation. These models permit a different attenuation index in each coordinate to fully capture the anisotropic nature of wave propagation in complex media. Analytical expressions that describe power law attenuation and anomalous dispersion in each direction are derived for these fractional calculus models.

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References

Meerschaert, M. M., Mortensen, J., and Wheatcraft, S. W., 2006, “Fractional Vector Calculus for Fractional Advection-Dispersion,” Phys. A, 367, pp. 181–190. [CrossRef]
Meerschaert, M. M., and Sikorskii, A., 2012, Stochastic Models for Fractional Calculus, De Gruyter, Berlin.
Samko, S. A., Kilbas, A., and Marichev, O., 1993, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London.
Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V., 2000, Fundamentals of Acoustics, 4th. ed., Wiley, New York, p. 119.
Gorenflo, R., Luchko, Yu., and Mainardi, F., 2000, “Wright Functions as Scale-Invariant Solutions of the Diffusion-Wave Equation,” J. Comput. Appl. Math., 118, pp. 175–191. [CrossRef]
Wear, K. A., 2001, “A Stratified Model to Predict Dispersion in Trabecular Bone,” IEEE Trans. Ultrason. Ferroelec. Freq. Control, 48(4), pp. 1079–1083. [CrossRef]
Nicholson, P. H. F., Haddaway, M. J., and Davie, M. W. J., 1994, “The Dependence of Ultrasonic Properties on Orientation in Human Vertebral Bone,” Phys. Med. Biol., 39(6), pp. 1013–1024. [CrossRef] [PubMed]
Anderson, C. C., Bauer, A. Q., Holland, M. R., Pakula, M., Wielki, K., Laugier, P., Bretthorst, G. L., and Miller, J. G., 2010, “Inverse Problems in Cancellous Bone: Estimation of the Ultrasonic Properties of Fast and Slow Waves Using Bayesian Probability Theory,” J. Acoust. Soc. Am., 128(5), pp. 2940–2948. [CrossRef] [PubMed]
Marutyan, K. R., Holland, M. R., and Miller, J. G., 2006, “Anomalous Negative Dispersion in Bone Can Result From the Interference of Fast and Slow Waves,” J. Acoust. Soc. Am., 120(5), pp. EL55–EL61. [CrossRef] [PubMed]
Haïat, G., Lhémery, A., Renaud, F., Padilla, F., Laugier, P., and Naili, S., 2008, “Velocity Dispersion in Trabecular Bone: Influence of Multiple Scattering and of Absorption,” J. Acoust. Soc. Am., 124(6), pp. 4047–4058. [CrossRef] [PubMed]
Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V., 2000, Fundamentals of Acoustics, 4th ed., Wiley, New York, pp. 211–212.
Chen, W., and Holm, S., 2004, “Fractional Laplacian Time-Space Models for Linear and Nonlinear Lossy Media Exhibiting Arbitrary Frequency Power-Law Dependency,” J. Acoust. Soc. Am., 115, pp. 1424–1430. [CrossRef] [PubMed]
Treeby, B. E., and Cox, B. T., 2010, “Modeling Power Law Absorption and Dispersion for Acoustic Propagation Using the Fractional Laplacian,” J. Acoust. Soc. Am., 127, pp. 2741–2748. [CrossRef] [PubMed]
Magin, R. L., Abdullah, O., Baleanu, D., and Zhou, X. J., 2008, “Anomalous Diffusion Expressed Through Fractional Order Differential Operators in the Bloch-Torrey Equation,” J. Magn. Reson., 190, pp. 255–270. [CrossRef] [PubMed]
GadElkarim, J. J., Magin, R. M., Meerschaert, M. M., Capuani, S., Palombo, M., Kumar, A., and Leow, A. D., 2013, “Directional Behavior of Anomalous Diffusion Expressed Through a Multidimensional Fractionalization of the Bloch-Torrey Equation,” Special Issue on Fractional-Order Circuits and Systems, IEEE J. Emerging Select. Topics Circuits Syst., 3(3), pp. 432–441. [CrossRef]
Kelly, J. F., McGough, R. J., and Meerschaert, M. M., 2008, “Time-Domain 3D Green's Functions for Power Law Media,” J. Acoust. Soc. Am., 124(5), pp. 2861–2872. [CrossRef] [PubMed]

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