Recently, Stiassnie et al. (J Ocean Eng Mar Energy 2(1):47–57, 2016) studied the potential for capturing wave energy over a large ocean basin via a toy model of a wave farm attacked by unidirectional wave fields. In the present work, we develop an approach to model macroscopically the behaviour of sparse arrays consisting of infinite rows of floating, axisymmetric wave energy converters in deep water. This approximate framework allows for such arrays to be characterized by frequency- and direction-dependent transfer functions. The example of a self-reacting converter consisting of vertically floating, coaxial cylinders moving in three modes of motion is discussed in detail, and the performance of large arrays of such devices, attacked by directional JONSWAP spectra and taking into account wave growth by the wind is investigated. This allows for fast and flexible estimates of power absorption by arrays as well as of their effects on the wave field.