For wave propagation over a finite array of cycloidal bars or trenches, an analytical solution of wave reflection in terms of Frobenius series to the modified mild-slope equation (MMSE) is given. First, the present solution related to cycloidal bars is validated against experimental data and two numerical solutions for wave reflection by rectified cosinoidal bars with the bar parameters same to those cycloidal bars, and the agreement among these solutions is quite good. Second, the present solution is compared with BEM (boundary element method) solutions to Laplace equation. It is found that the present solution is at least valid for those bars or trenches with the largest slope being not greater than 1:1. For wave reflection by cycloidal bars, the well-known phenomenon called phase downshift of Bragg resonance is confirmed again, which increases with increasing the bar height or width. For wave reflection by cycloidal trenches, it is revealed at first time that Bragg resonance always occurs at the phase with the trench distance being greater than the half incident wavelength. This new phenomenon is named as phase upshift in this paper. It is found that, the phase upshift of Bragg resonance increases with increasing the trench depth or width.

对于在有限摆线杆或沟槽阵列上的波传播，根据 Frobenius 级数对修正的温和斜率方程 (MMSE) 给出了波反射的解析解。首先，与摆线杆相关的当前解决方案通过实验数据和两个数值解进行了验证，这些解是由与那些摆线杆相同的杆参数的整流余弦杆反射波的数值解，这些解之间的一致性非常好。其次，将本解与拉普拉斯方程的 BEM（边界元法）解进行比较。发现本解决方案至少对那些最大坡度不大于1:1的条或沟有效。对于摆线棒的波反射，再次证实了众所周知的布拉格共振相位下移现象，它随着条形高度或宽度的增加而增加。对于摆线沟槽的波反射，首次揭示了布拉格共振总是发生在沟槽距离大于入射波长一半的相位。这种新现象在本文中称为相移。发现，随着沟槽深度或宽度的增加，布拉格共振的相位上移增加。