On harmonic plane wave propagation under fractional order thermoelasticity: an analysis of fractional order heat conduction equation

Abstract

In the present paper, we investigate the propagation of an harmonic plane wave propagating with assigned frequency by implementing the thermoelasticity theory based on a fractional order heat conduction law where the fractional order parameter (Formula presented.) satisfies (Formula presented.). After formulating the problem, the exact dispersion relation solutions for the plane wave are determined analytically and asymptotic expressions of different characterization of the wave are analyzed in two special cases, namely for a high-frequency field and low-frequency field. We consider the case of longitudinal wave which is coupled with the thermal field and we ignore the transverse wave as it is observed to be independent to the thermal field. Two different modes: predominantly thermal and predominantly elastic mode longitudinal waves are found. Finally we compute wave characterizations for the intermediate values of frequency and verify our analytical results for the limiting cases of wave frequency. A detailed analysis is presented to highlight the effects of the fractional order parameter, (Formula presented.), on the wave fields. Several important points are highlighted and the most important point which we have found is that (Formula presented.) plays a very important role in the behavior of the wave. As (Formula presented.), the thermal mode wave behaves similarly as in the case of a classical thermoelastic model and the nature of the wave changes significantly as (Formula presented.) gets nearer to the value 1, behaving more similarly to the case of a generalized thermoelastic model (Lord–Shulman model). © The Author(s) 2015.