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      Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional calculus

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          Abstract

          An analogy is drawn between the diffusion-wave equations derived from the fractional Kelvin-Voigt model and those obtained from Buckingham's grain-shearing (GS) model [J. Acoust. Soc. Am. 108, 2796-2815 (2000)] of wave propagation in saturated, unconsolidated granular materials. The material impulse response function from the GS model is found to be similar to the power-law memory kernel which is inherent in the framework of fractional calculus. The compressional wave equation and shear wave equation derived from the GS model turn out to be the Kelvin-Voigt fractional-derivative wave equation and the fractional diffusion-wave equation respectively. Also, a physical interpretation of the characteristic fractional-order present in the Kelvin-Voigt fractional derivative wave equation and time-fractional diffusion-wave equation is inferred from the GS model. The shear wave equation from the GS model predicts both diffusion and wave propagation in the fractional framework. The overall goal is intended to show that fractional calculus is not just a mathematical framework which can be used to curve-fit the complex behavior of materials, but rather it can be justified from real physical process of grain-shearing as well.

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          Author and article information

          Journal
          2015-12-15
          2015-12-22
          Article
          1512.05336
          0b83fb1e-7ccf-4212-8196-bd230bb10497

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          Preprint submitted to: The Journal of the Acoustical Society of America (JASA) on December 11, 2015. Total 38 pages and 7 EPS figures
          physics.geo-ph math-ph math.MP

          Mathematical physics,Geophysics,Mathematical & Computational physics
          Mathematical physics, Geophysics, Mathematical & Computational physics

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