This paper presents further development in a new time-domain computational technique developed for the determination of far-field acoustic radiation. This method, referred to as the transform potential-theoretic (TPT) method, solves linear time-dependent wave propagation problems in an unbounded medium, combining techniques involving numerical transforms/inverse transforms and potential theory. The end result is a robust procedure that is accurate and computationally efficient. The TPT technique is meshless, yet it can handle arbitrary geometries for the acoustic source. The validity of the procedure is demonstrated for the case of an acoustic wave propagation from a two-dimensional bounded surface embedded in a uniform flow. Both direct simulations as well as the current method(s) are compared. The single-layer potential theory based TPT method required fewer points on the boundary than the double-layer potential theory based TPT and performs better than direct simulation in high subsonic Mach number flows (M=0.8). The times required to compute these solutions were more than two orders of magnitude lower than direct simulation.