The generalized telegraph equation with moving harmonic source: Solvability using the integral decomposition technique and wave aspects
T. Pietrzak, A. Horzela, K. Górska
Abstract
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.