Design of Spherical Radiators Capable of Producing Prescribed Directivity Patterns

Abstract

Formulas are given that permit the calculation of the radial velocity at the surface of a spherical radiator, such that a prescribed pressure distribution will be achieved over the surface of any sphere, concentric with, and exterior to, the radiator. Formulas valid in the nearfield and the farfield are given. Both symmetrical and unsymmetrical pressure distributions are considered. A special class of symmetrical pressure distributions described by Chebyshev polynomials are discussed in detail. In particular, it is shown how these can be used to independently specify beam width and lobe suppression. Formulas and curves are also given for their directivity indices. A series of numerical cases were selected in which the beam width ranged from 10° through 90° and the side-lobe suppressions ranged from 10 through 40 dB. The distribution (continuous) of radiator surface velocity required to produce each of these was calculated on a high-speed digital computer. In addition, one of these continuous velocity distributions was approximated by discrete sets of radiating pistons having different sizes, and the pressure fields due to these arrays were compared to the pressure field that was originally specified. Some of the piston arrays reproduced the specified pressure pattern quite well.