This study explores the use of fractional partial differential equations to model the wave propagation through a one-dimensional complex and heterogeneous medium. In particular, this work discusses the use of fractional calculus to obtain closed form analytical solutions for the dispersion and propagation of elastic waves in a periodic, bi-material rod. From a mathematical standpoint, the approach allows converting a partial differential equation having spatially variable coefficients (i.e. the traditional wave equation in periodic media) to a space-fractional wave equation with constant coefficients. We show that the equivalent fractional equation exhibits a frequency-dependent and complex fractional order. Although this conversion might appear to increase the overall complexity of the model, in practice it enables obtaining closed form analytical solutions of wave propagation problems through inhomogeneous media.
The analytical solution to the space-fractional equation is obtained for the steady state response under harmonic loading. The result is compared to the traditional finite element solution of the wave equation in periodic media showing that the new approach is able to provide a reliable and accurate analytical representation of the dynamic response of the medium.
John P. Hollkamp and Fabio Semperlotti, "Analysis of dispersion and propagation properties in a periodic rod via fractional wave equation (Conference Presentation)," Proc. SPIE 10598, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2018, 105981Z (Presented at SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring: March 08, 2018; Published: 27 March 2018); https://doi.org/10.1117/12.2295621.5759121646001.
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