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

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+vt, 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.