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Multiple reflections of electromagnetic waves between two planes are studied, and the increase in directivity that results by placing a partially reflecting sheet in front of an antenna with a reflecting screen is investigated at a wavelength of 3.2 cm. The construction and performance of various models of such arrays is discussed. Thus, for example, a "reflex-cavity antenna" with an outer diameter of and an over-all length of only is described which has half-power beamwidths of and in the and planes, respectively, and a gain of approximately 14 db. It is shown that larger systems produce considerably greater directivity but that their efficiency is poor.