The Defect State In The Locally Resonant Phononic Crystal

In recent years, the propagation of acoustic wave or elastic wave in periodic compositematerials or structures (phononic crystal) have been deeply investigated. Because of the existof the phononic band gap, the propergation of elastic or acoustic wave in the crystal showssome special properties, based which enomous potential applications were suggested.According to the previous researchs, when the defects are brought in the phononic crystal, adefect state with acoustic or elastic wave localized near the defects can be obtained. Such kindof defect states, which appears in the compelete band gap as narrow pass band, can be used aswave filter or waveguide. However, the previous study suggested defect states are mostlyrealized in a phononic crystal based on Bragg scattering, for the phononic crystal based on thelocal resonant mechanism, the realization of a defect mode is hard and rarely reported. Thereason of which mainly comes from the fact that, in the latter structure, the gap openedusually in a low frequency region where the wavelength is much longer than the latticeconstant. It is well known that, in the long wavelength limit, a structural defect in real spacecan hardly break the dispersion of the wave behavior. In this paper, we will investigate themechanism for the defect state in the phononic crystal based on local resonant units.(1) We analyze in detail the interaction between resonant units. We find that, in the localresonant phononic crystal, a strong interaction between resonators can exist when the incidentwave’s frequency is around the resonant frequency of the scatterers. Because this kind ofinteraction is passed by the lower-order scattering wave rather than the high-order evanescentmode, it always shows a long-region character and can hardly be broken by the real-spacestructural disorder. However, a significant change of such kind of interaction can be obtainedby changing the resonant character of the resonator units, by which a defect state can beobtained. Based on the understanding, we show that properties of the defect states, such as thelocalization degree and the bandwidth, can be controlled by changing the resonant charactersof the defect resonators.(2) Based on the previous understanding, we suggest a new kind of subwavelengthwaveguide by which the longitudinal and transverse wave modes can be separated. As weknow, in the local resonant phononic crystal, the bandgap relys strongly on the resonantfrequency of the resonant unit, and the resonant frequencies for longnitudual and transversevibrations are usually the same when the resonant units are geometrically symmetric.However, the situation can be changed if the resoantors have different resoant frequencies for longnitudual and transverse vibrations. We show that the structure can be realized byintroducing an unsymmetric "defect" resonator.