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Theory of Constant Directivity Circular-Arc Line Arrays

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We develop the theory for a broadband constant-beamwidth transducer (CBT) formed by a 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 determined by the prescribed beam width and arc radius. We derive two shading functions, with cosine and Chebyshev polynomial forms, optimized to minimize this cutoff frequency and thereby extend constant-beamwidth behavior over the widest possible band. We illustrate the theory with simulations of magnitude responses, full-sphere radiation patterns and directivity index, for example designs with both wide- and narrow-beam radiation patterns.

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