Doppler effect described by the solutions of the Cattaneo telegraph equation
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O R I G I NA L PA P E R
Yuriy Povstenko
· Martin Ostoja-Starzewski
Doppler effect described by the solutions of the Cattaneo telegraph equation
Received: 25 September 2020 / Accepted: 15 October 2020 © The Author(s) 2020
Abstract The Cattaneo telegraph equation for temperature with moving time-harmonic source is studied on the line and the half-line domain. The Laplace and Fourier transforms are used. Expressions which show the wave fronts and elucidate the Doppler effect are obtained. Several particular cases of the considered problem including the heat conduction equation and the wave equation are investigated. The quasi-steady-state solutions are also examined for the case of non-moving time-harmonic source and time-harmonic boundary condition for temperature.
1 Introduction The classical parabolic diffusion equation and hyperbolic wave equation have their own characteristic features. For example, dissipation and infinite velocity of the disturbance propagation are inherent to the diffusion equation, while wave fronts, finite speeds of propagation and the Doppler effect [1–3] are peculiar to the solutions of the wave equation. The time-fractional diffusion-wave equation (see [4–6] and the references therein) ∂αT = aT , ∂t α
0 < α ≤ 2,
(1)
with the Caputo time-fractional derivative describes different important physical phenomena in bodies with complex internal structure, and interpolates between the diffusion equation when α = 1 and the wave equation when α = 2. Overall, Eq. (1) exhibits inherent features of both types of equations [6–8]. The term “diffusion-wave” is also used in another context. Ångström [9] was the first to investigate the standard parabolic diffusion equation (heat conduction equation) under time-harmonic (wave) impact. This pioneering study had aroused considerable interest of researchers and was even translated into English [10]. To describe this type of physical phenomenon the term “oscillatory diffusion” is used, in parallel with the term “diffusion-wave” . An extensive review of literature on this subject can be found in [11–13]. There are two possibilities to introduce oscillations into the parabolic diffusion equation. The first one consists in adding the harmonic source term [14,15]; the second one involves time-harmonic boundary conditions [16,17]. Often, in previous studies, the quasi-steady-state oscillations were investigated when the solution was Y. Povstenko (B) Faculty of Science and Technology, Jan Dlugosz University in Czestochowa, Armii Krajowej 13/15, 42-200 Czestochowa, Poland E-mail: [email protected] M. Ostoja-Starzewski Department of Mechanical Science and Engineering, Beckman Institute and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA E-mail: [email protected]
Y. Povstenko, M. Ostoja-Starzewski
represented as a product of a function of the spatial coordinates and time-harmonic term eiωt with the angular frequency ω. In 1948 Cattaneo [18] proposed the evolution equation for the heat flux ∂q
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