the generalization of structure factor for rods by polygon section in two-dimensional phononic crystals

the purpose of this paper is the generalization of structure factor for rods by polygon section in two dimensional phononic crystals. if we use the plane wave expansion method (pwe) for the propagation of acoustic waves in 2d phononic crystals, structure factor will be an important quantity. in order to confirm the obtained relations, we have calculated the band structure for xy and z vibration modes in 2d phononic crystals and the propagation of bulk acoustic waves (baw) are considered. in addition, the effect of sides’ number on the band structure and the complete band gaps width are investigated. phononic crystals studied in this paper are composites medium of a square lattice consisting of parallel nickel rods embedded in epoxy. the frequency is calculated by pwe in the condition of elastic rigidity to the solid inclusions. the results showed that, when the section of rods have 2n+2 (n is even) and 2n+1  by increasing sides number of the rod sections, the bands of xy mode shift to lower-frequency, the bands are smoother and the width of the band gap increases, but the band of z mode has not changed by n variations. moreover, when the section of rods have 2n+2 (n is odd) by increasing sides number of rod sections, the band structure of xy mode changes slightly and the width of the complete band gap is decreased. this confirms the effect of lattice symmetry on the complete band gap width. but, the band structure of z mode has not changed.