For a circular plane piston of radiusa, producing an ultrasonic beam of propagation constantk(or 2π/λ), theoretical expressions are derived for special values of the velocity potential at high frequencies. These apply forka⩾ 100 at distanceszfrom the source governed by (z/a)3⩾ka. The velocity potential averaged over a measurement circle equal to the piston is found, and also the potential at the rim of such a circle. The first quantity agrees well with results computed otherwise by Huntington, Emslie, and Hughes. Some remarks on wave front shapes are added. One conclusion is that the assumption of a collimated plane‐wave beam near such a source is not universally very good, although it has some approximate relevance.