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The far fields of resonant circular arrays of cylindrical dipoles with only one or two dipoles driven have been examined in a number of recent works. The discrete currents on such arrays can be thought of as a slow standing or traveling wave. In the present paper, the near fields of such arrays are studied. In order to better understand the somewhat unusual behavior of the near fields, we first discuss some simpler two-dimensional problems involving slow traveling current waves. We find that the associated fields possess certain surface-wave characteristics. We also find important differences from the case of linear arrays, which has been extensively studied in the past. Our discussions include methods to measure the efficiencies of our radiators, the extent of the near-field region, and the onset of the far-field region.