Sound Field Reproduction of a Virtual Source Inside a Loudspeaker Array With Minimal External Radiation

We derived an integral equation for reproducing the sound field of a virtual source inside an array of loudspeakers with reduced radiation to the outside. Reproduction of a sound field over a finite interior region inevitably generates sound waves that propagate outside the region. This undesirable radiation is reflected from walls and can induce artifacts in the interior region. In principle, the Kirchhoff-Helmholtz (KH) integral can be used to reproduce the interior sound field from an exterior virtual source without any external radiation. However, if there is a virtual source inside the array, the integral formula does not explicitly demonstrate how one can reproduce the sound field or minimize the external radiation. In this work, we derive an explicit formula for reproducing a sound field with minimal external radiation when a virtual source is located inside a loudspeaker array. The theory shows that external radiation can be effectively reduced without solving any inverse problem. The proposed formula follows the form of the KH integral and thus requires monopole and dipole sources. Although dipole sources are difficult to build in practice, the theory predicts that sound field reproduction with minimal external radiation is possible and that the room dependency of the sound field reproduction system can be decreased.