For a circular plane piston of radiusa, producing an ultrasonic beam with propagation constantk(or 2π/λ), an expression is derived for the velocity potential or the acoustic pressure, averaged with respect to magnitude and phase over a “measurement circle” equal in area to the piston and centered in the beam. The expression should be highly accurate forka⩾ 100, at distanceszfrom the source governed by(z/a)3 ⩾ ka. It agrees well with results computed, in another way, by Huntington, Emslie, and Hughes. The assumption that relatively near the source there is a collimated beam of plane waves is shown to be not very accurate; the averaged pressure falls off monotonically over all distances considered. The velocity potential at the rim of the “measurement circle” is also computed, and compared with the plane wave assumption.