We present a detailed theoretical study on the acoustic band structure of two-dimensional(2D)phononic crystal.The 2D phononic crystal with parallelogram lattice structure is considered to be formed by rigid solid rods embedded in air.For the circular rods,some of the extrema of the acoustic bands appear in the usual high-symmetry points and,in contrast,we find that some of them are located in other specific lines.For the case of elliptic rods,our results indicate that it is necessary to study the whole first Brillouin zone to obtain rightly the band structure and corresponding band gaps.Furthermore,we evaluate the first and second band gaps using the plane wave expansion method and find that these gaps can be tuned by adjusting the side lengths ratio R,inclined angleθand filling fraction F of the parallelogram lattice with circular rods.The results show that the largest value of the first band gap appears atθ=90°and F=0.7854.In contrast,the largest value of the second band gap is atθ=60°and F=0.9068.Our results indicate that the improvement of matching degree between scatterers and lattice pattern,rather than the reduction of structural symmetry,is mainly responsible for the enhancement of the band gaps in the 2D phononic crystal. We present a detailed theoretical study on the acoustic band structure of two-dimensional (2D) phononic crystal. The 2D phononic crystal with parallelogram lattice structure is considered to be formed by rigid solid rods embedded in air. For the circular rods, some of the extrema of the acoustic bands appear in the usual high-symmetry points and, in contrast, we find that some of them are located in other specific lines. For the case of elliptic rods, our results indicates that it is necessary to to study the whole first Brillouin zone to obtain rightly the band structure and corresponding band gaps.Furthermore, we evaluate the first and second band gaps using the plane wave expansion method and find that these gaps can be tuned by adjusting the side lengths ratio R, inclined angle θand filling fraction F of the parallelogram lattice with circular rods.The results show that the largest value of the first band gap appears atθ = 90 ° and F = 0.7854. contrast, the largest value of the second band gap is at θ = 60 ° and F = 0.9068.Our results that that the improvement of matching degree between scatterers and lattice pattern, rather than the reduction of structural symmetry, is mainly responsible for the enhancement of the band gaps in the 2D phononic crystal.