On the scattering of a disk source by a rigid sphere for directivity broadening

The purpose of this study is to model a circular planar loudspeaker placed near a spherical reflector to broaden its directivity pattern, which would otherwise become increasingly narrow at high frequencies. Through ray tracing, it seems intuitively feasible to thus create a virtual point source at very high frequencies, but we provide a more rigorous analysis to determine what will happen at intermediate frequencies where the wavelength is of a similar magnitude to the diameter of the disk or sphere. We show that a smoother off-axis response is obtained with a dipole pressure source, which does not obstruct the scattered sound, rather than a monopole velocity source. Hence, an electrostatic loudspeaker, for example, would be more suitable than a dynamic one. The sphere may also serve as a spherical approximation of a human head, in which case the loudspeaker would become an open headphone that is not sealed to the ear.