Eigenvalue problems for doubly periodic elastic structures and phononic band gaps

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