We consider the problem of elastic waves propagating in a two–dimensional array of circular cavities, taking rigorous account of coupling between shear and dilational waves. A technique, originally due to Rayleigh, is derived that involves an elegant identity between the singular and non–singular components of the stress fields in the array. This leads to an infinite linear system which can be truncated and solved in order to determine the complete structure of the propagating modes. Of particular interest is the possibility of exhibiting phononic band gaps, i.e. domains of frequency for which all propagating vibration in the material is suppressed.