Evanescent waves in multi-resonator metamaterials: Mechanisms and characteristic parameters of wave attenuation

The existence of phononic band gaps in multi-resonator acoustic metamaterials is a prominent feature of this class of artificial materials. In this work, non-propagating (evanescent) acoustic waves in different one-dimensional chains containing multi-resonators are studied within the framework of the novel proposed general approach. The wave dispersion laws in wide frequency ranges are used to calculate the size and to determine the location of band gaps. The attenuation coefficients γ for frequencies ω inside the band gaps are obtained in the quasi-stationary mode, and the dynamics of γ(ω) dependencies is studied in great detail. The physical origin of the appearance of gaps in the wave spectrum of several multi-vibrators is elucidated. The equations of motion were derived and solved analytically and numerically, given the specific initial conditions. It is shown that waves, excited at one chain edge, which in the quasi-stationary approach should decay exponentially, i.e., represent truly evanescent waves, actually experience decrease in their amplitudes much more slowly. Moreover, even for long chains with several hundred vibrators, the excited waves with frequencies inside the band gaps never completely decay and there is non-zero amplitude of vibration at the end of the chains. This means that in reality, the acoustic wave evanescence in such metamaterials is essentially always incomplete. However, the residual oscillations have much smaller amplitudes compared to the magnitude of the amplitude of the excitation signal.