Complex band structures and effective medium descriptions of periodic acoustic composite systems

We examine the complex band structures and effective medium descriptions of a periodic acoustic composite system. It is shown that if the system exhibits a negative velocity band, the assignment of the system as a phononic crystal or as a ``double negative'' metamaterial is unambiguous only in some cases. An example is given where the system properties can be tuned gradually and continuously from an acoustic metamaterial to a phononic crystal, and there is no sharp dividing line between these two regimes.