Quasistatic stopband in the spectrum of one-dimensional piezoelectric phononic crystal

Propagation of a longitudinal wave through the periodic structure composed of alternating elastic and piezoelectric layers is considered. The faces of each piezoelectric layer are electroded and connected via a circuit with the capacity $C$. It is shown that if $C<0$ then the Floquet-Bloch spectrum $\omega (K)$ in a certain range of negative $C$ may possess a quasistatic absolute stopband starting at $\omega =0$. Other unusual features of the spectrum occurring at certain fixed values of $C<0$ are the infinite group velocity of the first branch at the origin point $\omega =0$, $K=0$ and the flat bands $\omega =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$.