Transient waves produced by a circle expanding for finite time

We construct transient solution of the initial-value problem to the inhomogeneous wave equation. The source is distributed on a circle expanding with the velocity of wave perturbation (the velocity of light for the electromagnetic waves). The expanding time is finite. The circle begins its expansion at the initial moment of time and remains in the plane. We obtain the solution of wave equation in the cylindrical coordinate system by using the incomplete separation of variables, the Riemann formula, and the relation containing three Bessel functions. The obtained expressions differ from the traditional representations in terms of the spherical harmonics. We describe the peculiarities of the space-time structure of the waves. We compare the found results with expressions for the case of the source belonging to the circle on the expanding sphere