Analysis of the complex sound power in the near field of spherical loudspeaker arrays

Abstract

Over the past decade, compact spherical loudspeaker arrays have emerged as directivity controlled electroacoustic sources. The usual approach to control such a device is to provide it with a finite set of pre-programmed directivities corresponding to spherical harmonics or to the acoustic radiation modes (ARMs) of the array. However, these orthogonal bases are not suited for the acoustic control in the near field, which requires a number of spherical harmonics larger than the number of loudspeakers, and modes that take into account the reactive part of the complex sound power, not only the active part as is the case for the conventional ARMs. This paper presents a theoretical and numerical study of alternative ARM approaches applied to spherical loudspeaker arrays that consider both components of the sound power, with focus on the arrangements following the regularity of the five Platonic solids. For such arrays, it is shown that the active, reactive, and the so-called generalized ARMs form exactly the same basis, which does not depend either on frequency, source-receptor distance or on the vibration pattern and shape of the individual loudspeakers, provided that these are identical and axisymmetric. Closed-form expressions for the modal active and reactive efficiencies are derived and evaluated for the Platonic solid loudspeakers as a function of frequency and source-receptor distance. In addition, an application example in local active noise control is presented, in which a spherical loudspeaker array is used as a secondary source aiming at producing a zone of quiet in its near field.