Theory of continuously curved and phased line sources for sound reinforcement

To supply large audience areas uniformly with amplified direct sound, large-scale sound reinforcement often employs line-source loudspeaker arrays adapted to the listening area by either adjusting the angles or delays between their individual elements. This paper proposes a model for such or smaller line-source loudspeakers based on a delayed Green’s function integrated over an unknown contour. For a broad frequency range, stationary phase approximation yields a differential equation that we utilize to find a curve and delay progression providing direct sound levels rolling off with −6 β dB per doubling of the distance; curve and phase designs can also be mixed to meet simultaneous targets using multiple design parameters β . The effectiveness of the formalism is proven by simulations of coverage, directivity, and discretization artifacts. Measurements on a miniature line array prototype that targets medium-scale immersive sound reinforcement applications verify the proposed theory for curvature, delay, and mixed designs.