Integral Approach for Reproduction of Virtual Sound Source Surrounded by Loudspeaker Array

A method for reproducing a desired sound field by using an array of loudspeakers is proposed. A virtual sound source that generates the desired sound field can be located at either the outside or inside a loudspeaker array; however, a complete reproduction of the virtual source located inside the array is physically not possible because the sound field from the internal virtual source should satisfy the inhomogeneous wave equation. When time reversal is used for removing such inhomogeneity, convergent waves toward the location of the virtual source always exist. The removal of these artifacts is an objective of our study. Most of the theoretical forms for the suppression of artifacts have been derived from the approximated Kirchhoff-Helmholtz integral incorporating a curved or line array. In this paper, we aim to develop a method based on a complete three-dimensional integral formula, which does not require an inversion of sound fields. Using the formula, we can directly predict the behavior of artifacts and design the excitation of the loudspeakers that effectively suppresses the artifacts. This paper also highlights a single-layer formula that only incorporates monopole arrays for the reproduction of the virtual source inside them.