Theory of continuously curved and phased line sources for sound reinforcement

Abstract

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 −6β 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.