Equations developed by L. V. King [Can. J. Res.11, 135–155 (1934)] can lead to useful expressions for the velocity potential of a circular plane piston, in an infinite rigid baffle, radiating into a dissipationless fluid. An exact series is derived, in negative powers of distance from the center of the piston face to the field point, with specific formulas for all coefficients. The first term is the Fraunhofer approximation. It is then shown that a much simpler series, with the same first term and with explicit coefficients, closely approximates the exact one, provided that the piston radius is at least a few acoustic wavelengths in size and that the field point, as viewed from the source, is at only a small angle from the beam axis. This approximate series always converges, and rapidly so for distances from just outside the near field to the remote regions governed accurately by the Fraunhofer expression. At any distance outside the near field, the angular range of validity includes almost all of the power radiated in the beam.