A brief tutorial review of Constant Beamwidth Theory (CBT), as first developed by the military for underwater transducers (JASA, 1978 July and 1983 June), is described. In this paper the transducer is a circular spherical cap of arbitrary half-angle with Legendre function shading. This provides a constant beam pattern and directivity with extremely low side lobes for all frequencies above a certain cutoff frequency. This paper extends the theory by simulation to discrete-source loudspeaker arrays, including: 1) circular wedge line arrays of arbitrary sector angle, which provide controlled coverage in one plane only; 2) circular spherical caps of arbitrary half-angle, which provide controlled axially symmetric coverage; and 3) elliptical toroidal caps, which provide controlled coverage for arbitrary and independent vertical and horizontal angles.:
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