The generalized telegraph equation with moving harmonic source: Solvability using the integral decomposition technique and wave aspects

The paper is devoted to study the frequency shift in the solution of the generalized telegraph equation with a moving point-wise harmonic source. This equation contains the nonlocality in time derivatives which is expressed by the memory functions η(t) and γ(t), where η(t) smears the second time-derivative and γ(t) the first one. Moreover, in the Laplace domain we have ηˆ(s)=γˆ2(s). The generalized telegraph equation with an external source is solved by using the integral decomposition which allows us to write this solution as a product of the solution of the telegrapher equation with harmonic source and fγˆ(ξ,t) which is a function of the Laplace transform of memory function γ. Such obtained solution manifests the frequency shift which is illustrated in three examples of the memory functions γ(t): the localized case, its mixture with power-law, and the power-law case only. We show that only the first two cases have the wave front and the Doppler-like shift. The third example, despite the lack of wave fronts, also manifests the frequency shift. Thus it turns out that the frequency shift occurs regardless of the existence of a wave front, but it is more visible when such a front exists.