Interior and exterior sound field control using general two dimensional first-order sources

Reproduction of a given sound field interior to a circular loudspeaker array without producing an undesirable exterior sound field is an unsolved problem over a broadband of frequencies. At low frequencies, by implementing the Kirchhoff–Helmholtz integral using a circular discrete array of line source loudspeakers, a sound field can be recreated within the array and produce no exterior sound field, provided that the loudspeakers have azimuthal polar responses with variable first-order responses which are a combination of a two-dimensional (2D) monopole and a radially oriented 2D dipole. This paper examines the performance of circular discrete arrays of line-source loudspeakers which also include a tangential dipole, providing general variable-directivity responses in azimuth. It is shown that at low frequencies, the tangential dipoles are not required, but that near and above the Nyquist frequency, the tangential dipoles can both improve the interior accuracy and reduce the exterior sound field. The additional dipoles extend the useful range of the array by around an octave.