This report develops the theory for a constant-beamwidth transducer formed by an unbaffled, continuous circular-arc isophase line source. Appropriate amplitude shading of the source distribution leads to a far-field radiation pattern that is constant above a cutoff frequency. If the active part of the array is limited to an arc of 180° or less, the radiation pattern is asymptotically frequency-independent above a cutoff frequency determined by the arc radius and the highest-order nonnegligible shading mode. The cutoff frequency is inversely proportional to the arc radius and prescribed beam width. Two shading functions are derived with cosine and Chebyshev polynomial forms, optimized to minimize this cutoff frequency and thereby extend constant-beamwidth behavior over the widest possible band. The authors illustrate the theory with simulations of magnitude responses, full-sphere radiation patterns, and directivity index for example arrays with both wide- and narrow-beam radiation patterns. The theory is extended to describe the behavior of circular-arc arrays of discrete point sources. The conclusions that follow are remarkably parallel to those for an amplitude-shaded spherical cap as developed in previous research.
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