Expectation-based approach for one-dimensional randomly disordered phononic crystals

An expectation-based approach to the statistical theorem is proposed for the one-dimensional randomly disordered phononic crystal. In the proposed approach, the expectations of the random eigenstates of randomly disordered phononic crystals are investigated. In terms of the expectations of the random eigenstates, the wave propagation and localization phenomenon in the random phononic crystal could be understood in a statistical perspective. Using the proposed approach, it is proved that for a randomly disordered phononic crystal, the Bloch theorem holds in the perspective of expectation. A one-dimensional randomly disordered binary phononic crystal consisting of two materials with the random geometry size or random physical parameter is addressed by using the proposed approach. From the result, it can be observed that with the increase of the disorder degree, the localization of the expectations of the eigenstates is strengthened. The effect of the random disorder on the eigenstates at higher frequencies is more significant than that at lower frequencies. Furthermore, after introducing the random disorder into phononic crystals, some random divergent eigenstates are changed to localized eigenstates in expectation sense.