Bragg resonant reflection of surface waves from deep water to shallow water by a finite array of trapezoidal bars

For surface waves propagating over a finite array of trapezoidal artificial bars from deep water to shallow water, an analytical solution in terms of Taylor series to the modified mild-slope equation (MMSE) is constructed. It is shown that the reflection coefficient totally depends upon the bar number, N, and five dimensionless parameters with respect to the incident wavelength L, i.e., the bottom width of the bars, W, the top width of the bars, W5, the water depth, H-0 (or K-0 = 2 pi H-0), the bar submergence, H-1 (or K-1), and the bar spacing, D. It is revealed at first time that, when the four dimensionless parameters W, W-0, K-0 and K-1 are fixed, the reflection coefficient is a periodic function of 2D with the period being unit. Based on the present analytical solution, influence of the bar number, bar width and bar height on Bragg resonant reflection, including the peak value, peak phase, and resonance bandwidth, is intensively analyzed. It is found that the peak value of the primary Bragg resonant reflection is not always greater than the peak values of all the subharmonic Bragg resonant reflections. When the bar number is relatively big, the peak phase will remain unchanged even if the bar number increases. In addition, with the increase of the bar width or bar height, the peak phases of the primary and subharmonic Bragg resonances downward shift more to a low frequency.