  This study explores the use of fractional calculus as a possible tool to
model wave propagation in complex, heterogeneous media. We illustrate the
methodology by focusing on elastic wave propagation in a one-dimensional
periodic rod. The governing equations describing the wave propagation problem
in inhomogeneous systems typically consist of partial differential equations
with spatially varying coefficients. Even for very simple systems, these models
require numerical solutions which are computationally expensive and do not
provide the valuable insights associated with closed-form solutions. We will
show that fractional calculus can provide a powerful approach to develop
comprehensive mathematical models of inhomogeneous systems that can effectively
be regarded as homogenized models. Although at first glance the mathematics
might appear more complex, these fractional order models can allow the
derivation of closed-form analytical solutions that provide excellent
estimations of the systems' dynamic responses. Equally important, these
solutions are valid in a frequency range that goes largely beyond the
well-known homogenization limit of traditional integer order approaches,
therefore providing a possible route to high-frequency homogenization. More
specifically, this study focuses on the analyses of the dispersion and
propagation properties of a periodic medium under single-tone harmonic
excitation and illustrates the methodology to obtain a space-fractional wave
equation capable of capturing the behavior of the physical system. The
fractional wave equation and its analytical solution are compared with
numerical results obtained via a traditional finite element method in order to
assess their validity and evaluate their performance. It is found that the
resulting fractional differential models are, in their most general form, of
complex and frequency-dependent order.
