Telegraph equation in polar coordinates: Unbounded domain with moving time-harmonic source

The telegraph equation with moving time-harmonic source with angular frequency omega is considered in polar coordinates (r, phi). Two problems are studied: the source moving on a straight line with constant velocity v and the source traveling on a circumference of a circle of radius R with a constant orbital fre-quency Omega The solutions are obtained using the integral transforms technique. Two limiting cases of the telegraph equation are also analysed: the Fourier heat conduction equation and the linear wave equa-tion. The singularity of the solution to the wave equation at a point r = R + vt, phi = 0 in the case of the source moving on a straight line is described. The relationship between the orbital frequency Omega and the polar coordinates at the wave front has been analyzed. The results of numerical simulation are presented graphically for a wide spectrum of nondimensional parameters. (c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

This article investigates the telegraph equation with a moving time-harmonic source in polar coordinates (r, phi). Two scenarios are examined: a source moving on a straight line with constant velocity v and a source traveling on a circumference of a circle with a constant orbital frequency Omega. Solutions are derived using integral transforms technique. Additionally, the limiting cases of the telegraph equation, namely the Fourier heat conduction equation and the linear wave equation, are analyzed. The singularity of the wave equation solution at the point r = R + vt, phi = 0 for the straight-line motion scenario is described. The relationship between the orbital frequency Omega and the polar coordinates at the wave front is explored. Numerical simulation results are presented graphically for various nondimensional parameters.