(a) Dispersion relation for a periodic array of resonators placed symmetrically with respect to the wave propagation direction (x). A local resonance bandgap opens at the frequency Ω₀ due to the coupling between the propagating wave on the waveguide and the resonant mode of the resonators. The bandgap opens at the edge of the Brillouin Zone, with a relatively large zero-group velocity region. (b) Dispersion relation for aperiodic array with asymmetric resonators with respect to the wave propagation direction (x). In such a case, a zero-group velocity mode arise within the Brillouin Zone due to the coupling between flexural (z-polarized) and torsional (ϕ_z-polarized) waves with opposite group velocity. (c,d) Arrangement of resonators corresponding to the dispersion relations in (a) and (b) respectively.