The Band Gap Mechanism And Effective Parameters Of Locally Resonant Acoustic Metamaterials

In recent years, the propagation behaviors of acoustic or elastic waves in the artificial composite materials, such as phononic crystal or acoustic metamaterial, have attracted great attention. The acoustic band gap is the most significant characteristic of the composite materials. Due to the existence of the band gap, the elastic and acoustic waves propagating in the structure exhibit many novel physical phenomena, such as negative refraction, extraordinary transmission, cloaking for acoustic wave and so on. Due to the existence of local resonant units in the acoustic metamatertials, acoustic excitation with specific frequency makes the metamaterials have some negative responses. Therefore, the metamaterials show a great potential application. In this paper, firstly, by using a one-dimensional model, we analyzed the mechanism of the band gap mechanism for resonator-based metamaterial, a general analytical condition for band gap is obtained.Secondly, by taking a two-dimensional acoustic metamaterials as example, we presented a new numerical method for the effective parameter calculation. The paper is divided into two parts:(1) Taking a taut string with spring-mass resonators as example, we obtained the general condition for band gap resonator-based metamaterial structure.According to this condition, it can be found that the dispersion relation of the resonator-based metamaterial is in general a result of the scattering phase and propagating phase. The phenomenon that the band gap is less dependent on lattice structure appears only in the special system in which the coupling between the resonators and the host medium is weak enough. For strong coupled systems, the dispersion of wave can be significantly adjusted by the propagating phase. Based on the understanding, a general guide for band gap optimization is given and the mechanism for structures with the defect states at sub-wavelength scale is revealed.(2) Since the quasi-static approximation of the traditional effective medium theory can no longer be satisfied in metamaterial, the effective medium descriptions in previous works are in fact based on the assumption that the metamaterial is a discrete medium, which means the phase accumulation of the wave changes abruptly when it pass across a single unit. We showed that, because the wave in the metamaterial is in general the Bloch wave, the effective wave velocity of the metamaterial can be obtained directly by the dispersion relation of the structure. As for the effective impedance of the structure, we showed that, by constructing a "homogeneousmedium/acoustic metamaterial" interface system, and according to continuous condition on its interface, we can establish a relation between the inputting plane modes outside and the propagating Bloch modes inside the metamaterial, by which the effective impedance can be obtained. In the calculation, we found further that only the lowest order Bloch modein the structure is dominant, which means the effective impedance can be calculated directly from the eigen field. The validity of the results is verified by comparing the transmission and reflection spectra of the finite-thickness structure with the ones of the corresponded effective medium.