Spherical-harmonic representation of transient waves produced by sources on a superluminal expanding sphere

The goal of this report is to construct the explicit solutions to the inhomogeneous wave equation in terms of transient spherical harmonics. These solutions describe waves in free space formed by the source distribution on a sphere expanding with a velocity greater than the wavefront velocity. In the particular case we get a description of waves produced by the source distribution on the specific expanding circle that belongs to the sphere, and give the algorithm constructing solutions to the problem of wave formation by a point source moving on a circle. The results obtained allow us to write at once solutions of analogous problems in the case of a sphere expanding with a velocity equal and less than the wavefront velocity. The one-component radial Hertz vector, introduced by Debye (1909) and Bromwich (1919), permit us to apply the scalar wave functions to electromagnetic waves. Solutions of the initial-value problem are constructed with the help of the method described by Manankova (1972).