Generalized telegraph equation: frequency shift

by Katarzyna Górska (IFJ PAN, Krakow)

Thursday 11 Apr 2024, 14:00 → 15:00 Europe/Warsaw

4402 (IFJ PAN)

The seminar is devoted to the study of frequency shifts occurring in solutions of the generalized telegraph equation with a moving point-wise harmonic source. I start with observing the frequency shift in the solution of the (standard) telegraph equation. Then, using the integral decomposition method I derive the non-local telegraph equation called also the generalized one. This non-locality in time derivatives is expressed by the memory functions $\eta(t)$ and $\gamma(t)$, where $\eta(t)$ smears the second time-derivative and $\gamma(t)$ the first one. The solution is presented as a product of the solution of the (standard) telegraph equation with harmonic source and a function of the Laplace transform of memory function. Such obtained solution manifests the frequency shift which is illustrated in the following examples of the memory functions: the localized case which is related to the (standard) telegraph equation, its mixture with power-law, and the power-law case only. 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.