A method for obtaining a type of progressing waves is introduced. The method is applied to show that
(α being a constant) is a progressing wave satisfying the wave equation c2∇2φ = ∂2φ/∂t2 in cylindrical coordinates r, θ and z, for an arbitrary analytic function F of a complex variable. In terms of this and other similar progressing waves, we consider the problem of wave propagation from a moving point source in two semi-infinite fluid spaces. Both the subsonic and supersonic cases are included. The solutions for a fixed line source and for a stationary point source are obtained as limiting cases.