Transmission of elastic and acoustic waves propagating in phononic crystals analyzed by a boundary element method

The transmission properties of elastic and acoustic waves propagating in a two-dimensional mixed solid-fluid phononic crystal are studied by using the boundary element method (BEM). The phononic system is composed of a series of periodically spaced infinite solids embedded in a fluid matrix, in which the solid has finite periodic layers in one direction and are periodically spaced in another direction. Taking into account the periodicity of the scattered wave field, we adopt the periodic Green´s functions as the fundamental solutions, which satisfy the elastic wave/acoustic equations and the Bloch´s periodicity condition. Based on the periodic Green´s functions, the boundary integral equations (BIEs) are constructed. For convenience, we use the constant boundary elements to discretize these BIEs. By using the boundary and continuity conditions, the discretized BIEs are solved numerically to obtain the transmitted wave field. The relations between the transmission coefficient and the frequency are determined, from which the band gaps of the phonic system can be clearly recognized. To show the efficiency and the accuracy of the present BEM, numerical examples for a mixed solid-fluid phononic system composed of steel cylinders in the water are presented. The results calculated by the present BEM are compared with those obtained by other methods. The results obtained by these methods agree well. It is shown that the present BEM is an efficient numerical method to compute the transmission properties of elastic and acoustic waves in phononic crystals with mixed solid-fluid components.