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

TL;DR

This paper links grain-shearing mechanisms in marine sediments to fractional calculus, showing that wave propagation equations derived from physical models correspond to fractional differential equations, thus providing a physical basis for fractional calculus in this context.

Contribution

It establishes a physical connection between grain-shearing in sediments and fractional calculus, deriving wave equations that reflect real material behavior.

Findings

Wave equations from the grain-shearing model match fractional calculus equations.

The fractional order has a physical interpretation related to grain-shearing.

The model predicts both diffusion and wave propagation in sediments.

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.