  An innovative special class of tunable periodic metamaterials is designed,
suitable for realising high-performance acoustic filters. The metamaterial is
made up of a phononic crystal coupled to local resonators. Such local
resonators consist of masses enclosed into piezoelectric rings, shunted by
either dissipative or non-dissipative electrical circuit. By tuning the
impedance/admittance of such electrical circuits, it is possible to fully
adjust the constitutive properties of the shunting piezoelectric material. This
feature paves the way for unconventional behaviours, well beyond the
capabilities achievable with classical materials. It follows that the acoustic
properties of the periodic metamaterial can be adaptively modified, in turn,
opening new possibilities for the control of pass and stop bands. By exploiting
a generalization of the Floquet-Bloch theory, the in-plane free wave
propagation in the tunable metamaterial is investigated, by varying a certain
tuning parameter, to show the efficiency of the proposed shunting piezoelectric
system as a wave propagation control device. Particular attention is devoted to
the determination of the in-plane constitutive equations of the shunting
piezoelectric phase in the transformed Laplace space. Finally, broad design
directions of tunable acoustic filters aiming to a changing performance
requirement in real-time, is also provided.
