The cardioid pattern from a two-element array was reported back as least
as far as 1937, by the famous George H. Brown. In the ideal case (free
space, no losses, etc.) the radiation directly to the rear is precisely
zero.
If you add various real world effects then the back lobe is not
precisely zero, and this is shown in the ARRL Antenna Book referenced by
Cecil.
A detailed description of all of this is in Kraus' Antennas, 2nd Ed., in
Chapter 11. He uses equations, and he does not mention coordination of
balloon patterns, so perhaps you have another new invention.
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