Can the Doppler be useful for a benchmark analysis of the wave-like properties of memory-dependent telegraphers’ equation

Telegraphers’ equation perturbed by a uniformly moving external harmonic impact is investigated to uncover information useful for distinguishing properties of the time evolution patterns that describe either memoryless or memory-dependent modeling of transport phenomena. Memory effects are incorporated into telegraphers’ equation by smearing the first- and second-order time derivatives so that the memory kernel smearing the second-order time derivative acts as the smeared derivative of the smeared first-order time derivative. Such a generalized telegraphers’ equation (abbreviated as GTE) is solved under initial conditions that specify the values of the solutions and their time derivatives taken at the initial time and boundary conditions that require the sought solutions to vanish either at the $x$ space infinity or the $(+l)/(−l)$ boundaries of a compact domain. The question is which solutions would be classified as traveling or standing waves. To answer this, we consider the Doppler effect and investigate how the frequency and velocity of external sources influence the obtained solutions. Using the short-time Fourier transform allows us to advance the problem and shows that infinite domain solutions to the GTEs, provided by a model example involving the Caputo fractional derivatives $C D t 2 α$ and $C D t α$ with $0<α≤1$⁠, exhibit a kind of velocity-dependent Doppler-like frequency shift if $1 2<α≤1$⁠. The effect remains unnoticed if $0<α≤ 1 2$⁠. This confirms our previous hypothesis that the emergence of wave-like effects in solutions of fractional equations is related to the occurrence of fractional time derivatives of the order greater than 1.