A class of cylindrically symmetric exact solutions of the wave equation

A class of analytically exact solutions is derived of the two-dimensional wave equation. The solutions are due to a point source (i.e. a uniform line source in 3D) at the origin r=0 with polynomial time dependence that starts at t=0. They are essentially similarity solutions in the variable c₀t/r. Their explicit formulation is supported by analytical results for large t and small r behaviour. By taking advantage of linearity, sources of finite duration can be modelled by subtracting from a solution its delayed version. For illustration, some examples are given graphically as functions of r, of t, and of t−r/c₀, for a number of points in time and location, respectively.