The theory of a broadband constant beamwidth transducer which is to be used primarily as a projector is presented. The transducer is a spherical cap of arbitrary half angle α shaded so that the normal velocity is equal toU0Pν(cos ϑ), wherePνis the Legendre function whose root of smallest angle occurs at ϑ=α. The required value for ν, the order of the Legendre function (which is not, in general, an integer) can be obtained to within 1% for α?1 radian from the approximation ν?0.5[(4.81/α)−1]. The transducer is shown to have uniform acoustic loading, extremely low sidelobes, and an essentially constant beam pattern for all frequencies above a certain cutoff frequency. Under piezoelectric drive the transducer is shown to have a flat transmitting current response over a broad band.