Transmission spectra of two-dimensional liquid-liquid phononic crystals

A boundary element method (BEM) is presented to compute the transmission spectra of two-dimensional (2D) liquid-liquid phononic crystals. The systems are composed of square or triangular lattices which have finite layers along the x-direction and are infinite along y-direction. The cross sections of the inclusions may be arbitrary. In a periodic unit cell, boundary integral equations of the matrix and the inclusion are given. Substituting the periodic boundary conditions and the interface conditions, a set of linear equations is formed, from the solutions of which the transmitted wave field can be obtained. Then the relations between the transmission spectra and the frequency are determined, which clearly exhibit the band gaps of the phononic system. The results are consistent with the band structures of the corresponding infinite systems. Different numerical examples are used to illustrate the accuracy and efficiency of the boundary element method.