On the sound fields of oblate and prolate hemispheroids in infinite baffles for directivity control

The hemispheroid is presented as an apodized form for controlling the beam width of a sound source by varying its height-to-radius ratio. Directivity patterns, on-axis responses, and radiation impedances are calculated for various height-radius ratios of the oblate hemispheroid using spheroidal wave functions. It turns out that, for smaller angles at least, there is a direct relationship between the internal angle of the semi-elliptic cross section and the half-cone angle within which the far-field pressure is largely contained at high frequencies. The hemispheroid is compared with both a spherical cap, which produces a much less regular response, and a high-frequency asymptotic approximation. The high-frequency asymptotic approximation is in the form of a flat circular radiator with a delay that increases radially from the center to the perimeter, as used in some electrostatic loudspeakers. With the almost complete absence of lobes, this appears to be an effective alternative means of apodization to a shaded array and is more efficient because, unlike a shaded array, constant axial velocity is maintained over the whole surface. A high-frequency approximation is also derived for a prolate hemispheroid. Since this may be formed from a planar array, a beam steering option is added.