Superdirective spherical radiator

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Abstract
The condition for the maximum directivity from a spherical radiator is presented and it is shown that this directivity index is 20 log N, where N is the number of consecutive modes excited. For superdirectivity, the relationship among the expansion coefficients is particularly simple and a closed-form expression for the beam-pattern function results. Using this method, a high directivity may be obtained no matter how small the sphere is, provided that the prescribed velocity distribution is maintained. Examples of the resulting beam patterns are given.

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