Telegraph equation in polar coordinates: Unbounded domain with moving time-harmonic source. (15th June 2023) - Povstenko, Yuriy; Ostoja-Starzewski, Martin; Kyrylych, Tamara
Highlights: The telegraph equation with moving time-harmonic source is considered in polar coordinates. Two problems are studied: the source moving on a straight line with constant velocity and the source traveling on a circumference of a circle with a constant orbital frequency. The relationship between the orbital frequency 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. Abstract: The telegraph equation with moving time-harmonic source with angular frequency ω is considered in polar coordinates ( r, φ ) . 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 frequency Ω . 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 equation. The singularity of the solution to the wave equation at a point r = R + v t, φ = 0 in the case of the source moving on a straight line is described. The relationship between the orbital frequency Ω 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.