Band gaps for Bloch waves over an infinite array of trapezoidal bars and triangular bars in shallow water

The problem of shallow-water wave propagation over an infinite array of periodic trapezoidal bars and triangular bars is studied. By employing the linear shallow-water wave theory and the well-known Bloch theorem, the eigenvalue problem in terms of the wave number for given wave frequency, water depth, and geometric properties of the bars is formed. A closed-form solution of the eigenvalue problem is derived which breaks the restriction of piecewise constant water depth required in almost all of the previous analytic solutions. The solution is verified against the existing solution for the special case of rectangular bars. Based on the present solution, gap maps in various cases are plotted which exactly show the distribution of band gaps for wave propagation over both trapezoidal bars and triangular bars, and the influence of the given wave frequency and the geometric parameters of bars such as height and width on the occurrence of band gaps is analyzed. By using the gap maps presented in the paper, the condition under which the waves can be completely blocked by an infinite array of trapezoidal bars and triangular bars can be easily and exactly determined.