The acoustic beam of a circular plane piston (radiusa) is analyzed as it approaches either the short‐wavelength limit (SWL) or the large‐radius limit (LRL). Using a new expression for the beam, we show that with no absorption the SWL would be a cylindrically collimated beam, with radiusa, of plane waves. If absorptive effects are minor, the chief deviations during approach to the SWL occur in two decreasing volumes, a horn‐shaped paraxial region and a cylindrical shell of mean radiusa. One consequence is that any second harmonic, self‐generated in the beam, approaches its own SWL more rapidly than does the first harmonic. Approach to the LRL yields a known paradox: Even the piston's nearfield need not approach a uniform plane wave. We show that the presence of absorption does make the LRL a plane wave and that in the absence of absorption there is a valid reason for the paradox: The piston beam must have a unique axis of cylindrical symmetry; an unbounded plane wave cannot.