Zero Reflections of Linear Water Waves Crossing a Finite Periodic Array of Quasi-Idealized Artificial Bars
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摘要:
该文研究水波越过平整海床上N个p次拟理想人工沙坝组成的周期阵列时出现的零反射(亦即完全透射). 所谓p次拟理想沙坝，是指沙坝上方的水深函数为一个常数加上p次单项式，其中p为正整数. 研究表明，若沙坝关于最深水深的相对高度远小于1，则水波越过p=1的拟理想沙坝(即三角形)周期阵列时产生遗传性零反射的条件是沙坝宽度正好为入射波半波长的正偶数倍. 随着p增加，水波越过p次拟理想沙坝周期阵列时产生遗传性零反射的相位向低频移动. 当p趋于无穷，p次拟理想沙坝退化为矩形沙坝，此时产生遗传性零反射的条件是沙坝宽度正好为入射波半波长的正整数倍. 此外，任意相邻Bragg共振峰之间共生性零反射的个数为N-1，且这N-1个零反射恰好为第二类Chebyshev多项式UN-1(cos(πx))的全部零点. 若沙坝关于水深的相对高度不是很小，则相邻Bragg共振峰之间共生性零反射的个数仍为N-1，且这些零反射的相位近似等于UN-1(cos(πx))的N-1个零点再减去前后两个共振峰相位下移量的平均值，而后者可通过修正Bragg原理估算. 但对于遗传性零反射的相位，目前仍无有效办法进行预测. 无疑，本研究丰富了对海床上周期排列的人工沙坝激发的Bragg共振反射现象的理解，并在海岸保护和波浪能提取等方面具有潜在应用价值.
Abstract:The zero reflection of linear water waves when they pass over a finite periodic array of quasi-idealized bars of degree p on a flat seabed is studied. The so-called quasi-idealized bar of degree p refers to the water depth function above the bar is a constant plus a monomial of degree p, where p is a positive integer. The results show that, when water waves crossing a periodic array of quasi-idealized bars of degree 1 (i.e., triangular bars) with the relative bar height with respect to the water depth being much less than 1, the condition for the generation of genetic zero reflection is that the bar width is exactly a positive even multiple the half wavelength of the incident wave. As p increases, the phase of the genetic zero reflection shifts towards lower frequencies. When p approaches infinity, the quasi-idealized bars of degree p tend to be a rectangular bar, and the condition for genetic zero reflection is that the bar width decreases to a positive integer multiple of the half wavelength of the incident wave. In addition, the total number of symbiotic zero reflections between any adjacent Bragg resonance peaks is N-1, and the excitation condition for these zero reflections is that the ratio of the bar spacing to the half wavelength of the incident wave is exactly N-1 zero points of the Chebyshev polynomial of the second kind, UN-1(cos(πx)). If the relative bar height with respect to the water depth is not very small, the total number of symbiotic zero reflections between adjacent Bragg resonance peaks is still N-1, and the phases of these zero reflections are approximately equal to the N-1 zero points of UN-1(cos(πx)) minus the mean of the phase shift of the two adjacent resonance peaks, where the latter can be estimated by the modified Bragg's law. However, at present there is no effective method to predict the phase of the genetic zero reflection. Undoubtedly, this study enriches the understanding of the Bragg resonance reflection induced by periodically arranged artificial sandbars on the seabed, and has potential application values in coastal protection and wave energy extraction.
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Key words:
- quasi-idealized artificial bar /
- Bragg resonance reflection /
- genetic zero reflection /
- symbiotic zero reflection /
- Chebyshev polynomial of the second kind /
- modified Bragg's law
edited-byedited-by1) （我刊编委刘焕文来稿） -
图 1 p次拟理想人工沙坝周期阵列(p=1, 2, 3, 4, 10, ∞)
Figure 1. Periodic arrays of quasi-idealized bars of degree p=1, 2, 3, 4, 10, ∞
图 2 相对高度$ B \ll 1$时，两个拟理想沙坝激发的零反射, p=1, 2, 3, 4
Figure 2. The zero reflections caused by 2 quasi-idealized bars for $ B \ll 1$, p=1, 2, 3, 4
图 3 相对高度$ B \ll 1$时，三个拟理想沙坝激发的零反射, p=1, 2, 3, 4
Figure 3. The zero reflections caused by 3 quasi-idealized bars for $ B \ll 1$, p=1, 2, 3, 4
图 4 B不是很小时，p=1次拟理想沙坝(即三角形)阵列激发的零反射
Figure 4. The zero reflections caused by triangular bars with not-very-small B
图 5 N=4，相对高度B不是很小时, p=3次拟理想沙坝阵列激发的零反射
Figure 5. The zero reflections caused by quasi-idealized bars of p=3 with not-very-small B, N=4
图 6 相对高度$ B \ll 1$时，p=+∞次拟理想沙坝阵列激发的零反射, N=2, 3
Figure 6. The zero reflections caused by quasi-idealized bars of p=+∞ with $ B \ll 1$, N=2, 3
图 7 相对高度不是很小的矩形沙坝周期列引起的零反射及预测
注 为了解释图中的颜色，读者可以参考本文的电子网页版本.
Figure 7. The zero reflections and predictions caused by rectangular bars with not-very-small heights
表 1 $ B \ll 1$时，四类拟理想沙坝场激发的两类零反射
Table 1. The 2 types of zero reflections excited by 4 types of quasi-idealized bar fields for $ B \ll 1$
p type-1 (genetic) zero reflection type-2 (symbiotic) zero reflection p=1 $ \begin{gathered} W=1.0000, 2.0000, 3.0000, \cdots \\ \text { i.e., } 2 D=\frac{2.0000}{f}, \frac{4.0000}{f}, \frac{6.0000}{f}, \cdots \end{gathered}$ $ 2 D=n-1+\frac{j}{N}, n=1, 2, \cdots, j=1, \cdots, N-1$ p=2 $ \begin{gathered} W=0.7152, 1.2295, 1.7355, \cdots \\ \text { i.e., } 2 D=\frac{1.4304}{f}, \frac{2.4590}{f}, \frac{3.4710}{f}, \cdots \end{gathered}$ $ 2 D=n-1+\frac{j}{N}, n=1, 2, \cdots, j=1, \cdots, N-1$ p=3 $ \begin{gathered} W=0.6502, 1.2133, 1.7176, \cdots \\ \text { i.e., } 2 D=\frac{1.3004}{f}, \frac{2.4266}{f}, \frac{3.4352}{f}, \cdots \end{gathered}$ $ 2 D=n-1+\frac{j}{N}, n=1, 2, \cdots, j=1, \cdots, N-1$ p=4 $ \begin{gathered} W=0.6160, 1.1846, 1.7050, \cdots \\ \text { i.e. }, 2 D=\frac{1.2320}{f}, \frac{2.3692}{f}, \frac{3.4100}{f}, \cdots \end{gathered}$ $ 2 D=n-1+\frac{j}{N}, n=1, 2, \cdots, j=1, \cdots, N-1$ 
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表 2 p=1时，基于式(20)—(23)对第1、2和3阶共振峰值相位的预测
Table 2. Predicted phases of the 1st, 2nd and 3rd resonances based on eqs.(20)—(23) for p=1
number of iteration times D1XL D2XL D3XL 0 D1Bragg=1.0 D2Bragg=2.0 D3Bragg=3.0 1 0.965 5 1.931 6 2.899 1 2 0.965 5 1.931 6 2.898 9 3 2.898 9
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表 3 p=1, N=3时，式(25)对共生性零反射相位进行预测的相对误差
Table 3. Relative errors of predicted phases of the symbiotic zero reflections for N=3 and p=1
predicted phase actual phase relative error/% 0.316 1 0.319 6 1.10 0.649 1 0.644 2 0.76 1.281 9 1.288 4 0.50 1.615 2 1.608 0 0.45 2.248 6 2.257 2 0.38 2.581 9 2.576 8 0.20
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表 4 p=3时，基于式(20)—(23)对第1、2和3阶共振峰值相位的预测
Table 4. Predicted phases of the 1st, 2nd and 3rd resonances based on eqs.(20)—(23) for p=3
w=1 m w=1.5 m D1XL D2XL D3XL D1XL D2XL D3XL 0 1.0 2.0 3.0 1.0 2.0 3.0 1 0.952 3 1.956 0 2.976 9 0.933 8 1.940 2 2.972 3 2 0.951 0 1.954 2 2.976 3 0.931 3 1.936 6 2.971 3 3 0.951 0 1.954 1 2.976 3 0.931 2 1.936 3 2.971 2 4 1.954 1 0.931 2 1.936 3 2.971 2
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表 5 p=+∞时，沙坝阵列导致的前三阶共振峰值相位的预测
Table 5. Predicted peak phases of 3 Bragg resonances caused by bars with p=+∞
number of iteration times D1XL D2XL D3XL 0 D1Bragg=1.0 D2Bragg=2.0 D3Bragg=3.0 1 0.928 43 1.858 21 2.790 65 2 0.928 40 1.857 97 2.789 85 3 0.928 40 1.857 97 2.789 85
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表 6 p=+∞时，共生性零反射相位预测值的相对误差
Table 6. Relative errors of predicted phases of symbiotic zero reflections for p=+∞
predicted phase actual phase relative error/% 0.297 5 0.309 0 3.72 0.630 9 0.618 0 2.09 1.226 5 1.240 9 1.16 1.559 9 1.549 9 0.65 2.157 2 2.162 8 0.26 2.490 6 2.471 8 0.76
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[1] NEWMAN J N. Propagation of water waves past long two-dimensional obstacles[J]. Journal of Fluid Mechanics, 1965, 23(1): 23. doi: 10.1017/S0022112065001210 [2] MEI C C, BLACK J L. Scattering of surface waves by rectangular obstacles in waters of finite depth[J]. Journal of Fluid Mechanics, 1969, 38: 499-511. doi: 10.1017/S0022112069000309 [3] MEI C C. The Applied Dynamics of Ocean Surface Waves[M]. World Scientific, 1989. [4] LIN P, LIU H W. Analytical study of linear long-wave reflection by a two-dimensional obstacle of general trapezoidal shape[J]. Journal of Engineering Mechanics, 2005, 131: 822-830. [5] XIE J J, LIU H W, LIN P. Analytical solution for long-wave reflection by a rectangular obstacle with two scour trenches[J]. Journal of Engineering Mechanics, 2011, 137(12): 919-930. doi: 10.1061/(ASCE)EM.1943-7889.0000293 [6] LIU H W, LUO J X, LIN P, et al. Analytical solution for long-wave reflection by a general breakwater or trench with curvilinear slopes[J]. Journal of Engineering Mechanics, 2013, 139(2): 229-245. doi: 10.1061/(ASCE)EM.1943-7889.0000483 [7] CHAKRABORTY R, MANDAL B N. Water wave scattering by a rectangular trench[J]. Journal of Engineering Mathematics, 2014, 89(1): 101-112. doi: 10.1007/s10665-014-9705-6 [8] MEDINA-RODRÍGUEZ A, BAUTISTA E, MÉNDEZ F. Asymptotic analysis of the interaction between linear long waves and a submerged floating breakwater of wavy surfaces[J]. Applied Ocean Research, 2016, 59: 345-365. doi: 10.1016/j.apor.2016.06.002 [9] KAR P, KOLEY S, SAHOO T. Scattering of surface gravity waves over a pair of trenches[J]. Applied Mathematical Modelling, 2018, 62: 303-320. doi: 10.1016/j.apm.2018.06.002 [10] KAUR A, MARTHA S C, CHAKRABARTI A. Solution of the problem of propagation of water waves over a pair of asymmetrical rectangular trenches[J]. Applied Ocean Research, 2019, 93: 101946. doi: 10.1016/j.apor.2019.101946 [11] GOYAL D, MARTHA S C. Optimal allocation of multiple rigid bars with unequal draft to enhance Bragg reflection of surface gravity waves[J/OL]. Ships Offshore Structures, 2025[2025-06-12]. https://doi.org/10.1080/17445302.2025.2468001 .[12] BRAGG W H, BRAGG W L. The reflection of X-rays by crystal[J]. Mathematical, Physical and Engineering Sciences, 1913, 88: 428-438. [13] DAVIES A G. The reflection of wave energy by undulations on the seabed[J]. Dynamics of Atmospheres and Oceans, 1982, 6(4): 207-232. doi: 10.1016/0377-0265(82)90029-X [14] HEATHERSHAW A D. Seabed-wave resonance and sand bar growth[J]. Nature, 1982, 296: 343-345. doi: 10.1038/296343a0 [15] DAVIES A G, HEATHERSHAW A D. Surface-wave propagation over sinusoidally varying topography[J]. Journal of Fluid Mechanics, 1984, 144(1): 419-443. [16] LIU H W, LI X F, LIN P. Analytical study of Bragg resonance by singly periodic sinusoidal ripples based on the modified mild-slope equation[J]. Coastal Engineering, 2019, 150: 121-134. doi: 10.1016/j.coastaleng.2019.04.015 [17] PENG J, TAO A, LIU Y, et al. A laboratory study of class Ⅲ Bragg resonance of gravity surface waves by periodic beds[J]. Physics of Fluids, 2019, 31(6): 067110. doi: 10.1063/1.5083790 [18] LIANG B, GE H, ZHANG L, et al. Wave resonant scattering mechanism of sinusoidal seabed elucidated by Mathieu instability theorem[J]. Ocean Engineering, 2020, 218: 108238. doi: 10.1016/j.oceaneng.2020.108238 [19] GAO J, MA X, DONG G, et al. Investigation on the effects of Bragg reflection on harbor oscillations[J]. Coastal Engineering, 2021, 170: 103977. doi: 10.1016/j.coastaleng.2021.103977 [20] ZHANG H, TAO A, TU J, et al. The focusing waves induced by Bragg resonance with V-shaped undulating bottom[J]. Journal of Marine Science and Engineering, 2021, 9(7): 708. doi: 10.3390/jmse9070708 [21] NING D Z, ZHANG S B, CHEN L F, et al. Nonlinear Bragg scattering of surface waves over a two-dimensional periodic structure[J]. Journal of Fluid Mechanics, 2022, 946: A25. doi: 10.1017/jfm.2022.609 [22] HAO J, LI J, LIU S, et al. Wave amplification caused by Bragg resonance on parabolic-type topography[J]. Ocean Engineering, 2022, 244: 110442. doi: 10.1016/j.oceaneng.2021.110442 [23] MEI C C, HARA T, NACIRI M. Note on Bragg scattering of water waves by parallel bars on the seabed[J]. Journal of Fluid Mechanics, 1988, 186: 147-162. doi: 10.1017/S0022112088000084 [24] GUAZZELLI E, REY V, BELZONS M. Higher-order Bragg reflection of gravity surface waves by periodic beds[J]. Journal of Fluid Mechanics, 1992, 245: 301-317. doi: 10.1017/S0022112092000478 [25] LIU Y, YUE D K. On generalized Bragg scattering of surface waves by bottom ripples[J]. Journal of Fluid Mechanics, 1998, 356(1): 297-326. [26] MADSEN P A, FUHRMAN D R, WANG B. A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry[J]. Coastal Engineering, 2006, 53(5/6): 487-504. [27] CHANG H K, LIOU J C. Long wave reflection from submerged trapezoidal breakwaters[J]. Ocean Engineering, 2007, 34(1): 185-191. doi: 10.1016/j.oceaneng.2005.11.017 [28] PORTER R, PORTER D. Scattered and free waves over periodic beds[J]. Journal of Fluid Mechanics, 2003, 483: 129-163. doi: 10.1017/S0022112003004208 [29] LIU H W, LUO H, ZENG H D. Optimal collocation of three kinds of Bragg breakwaters for Bragg resonant reflection by long waves[J]. Journal of Waterway, Port, Coastal, and Ocean Engineering, 2015, 141(3): 1-17. [30] LIU H W, SHI Y P, CAO D Q. Optimization of parabolic bars for maximum Bragg resonant reflection of long waves[J]. Journal of Hydrodynamics (Series B), 2015, 27(3): 373-382. doi: 10.1016/S1001-6058(15)60495-4 [31] LIU H W, ZENG H D, HUANG H D. Bragg resonant reflection of surface waves from deep water to shallow water by a finite array of trapezoidal bars[J]. Applied Ocean Research, 2020, 94: 101976. doi: 10.1016/j.apor.2019.101976 [32] XIE J J. Long wave reflection by an array of submerged trapezoidal breakwaters on a sloping seabed[J]. Ocean Engineering, 2022, 252: 111138. doi: 10.1016/j.oceaneng.2022.111138 [33] GUO F C, LIU H W, PAN J J. Phase downshift or upshift of Bragg resonance for water wave reflection by an array of cycloidal bars or trenches[J]. Wave Motion, 2021, 106: 102794. doi: 10.1016/j.wavemoti.2021.102794 [34] 潘俊杰, 刘焕文, 李长江. 周期排列抛物形系列沟槽引起的线性水波Bragg共振及共振相位上移[J]. 应用数学和力学, 2022, 43(3): 237-254. doi: 10.21656/1000-0887.420123 PAN Junjie, LIU Huanwen, LI Changjiang. Bragg resonance and phase upshift of linear water waves excited by a finite periodic array of parabolic trenches[J]. Applied Mathematics and Mechanics, 2022, 43(3): 237-254. (in Chinese) doi: 10.21656/1000-0887.420123 [35] LIU H W, XIONG W, XIE J J. A new law of class Ⅰ Bragg resonance for linear long waves over arrays of parabolic trenches or rectified cosinoidal trenches[J]. Ocean Engineering, 2023, 281: 114973. doi: 10.1016/j.oceaneng.2023.114973 [36] XIE J J, LIU H W. Analytical study of Bragg resonances by a finite periodic array of congruent trapezoidal bars or trenches on a sloping seabed[J]. Applied Mathematical Modelling, 2023, 119: 717-735. doi: 10.1016/j.apm.2023.03.010 [37] LINTON C M. Water waves over arrays of horizontal cylinders: band gaps and Bragg resonance[J]. Journal of Fluid Mechanics, 2011, 670: 504-526. doi: 10.1017/S0022112010005471 [38] LIU H W, LIU Y, LIN P. Bloch band gap of shallow-water waves over infinite arrays of parabolic bars and rectified cosinoidal bars and Bragg resonance over finite arrays of bars[J]. Ocean Engineering, 2019, 188: 106235. doi: 10.1016/j.oceaneng.2019.106235 [39] LIU H W. Band gaps for Bloch waves over an infinite array of trapezoidal bars and triangular bars in shallow water[J]. Ocean Engineering, 2017, 130: 72-82. doi: 10.1016/j.oceaneng.2016.11.056 [40] LIU H W. An approximate law of class Ⅰ Bragg resonance of linear shallow-water waves excited by five types of artificial bars[J]. Ocean Engineering, 2023, 267: 113245. doi: 10.1016/j.oceaneng.2022.113245 [41] PENG J, TAO A F, FAN J, et al. On the downshift of wave frequency for Bragg resonance[J]. China Ocean Engineering, 2022, 36(1): 76-85. doi: 10.1007/s13344-022-0006-y [42] FANG H, TANG L, LIN P. Bragg scattering of nonlinear surface waves by sinusoidal sandbars[J]. Journal of Fluid Mechanics, 2024, 979: A13. doi: 10.1017/jfm.2023.1005 [43] FANG H, TANG L, LIN P. Theoretical study on the downshift of class Ⅱ Bragg resonance[J]. Physics of Fluids, 2024, 36(1): 017103. doi: 10.1063/5.0178754 [44] REY V, GUAZZELLIÉ, MEI C C. Resonant reflection of surface gravity waves by one-dimensional doubly sinusoidal beds[J]. Physics of Fluids, 1996, 8(6): 1525-1530. doi: 10.1063/1.868928 [45] DING Y, LIU H W, LIN P. Quantitative expression of the modified Bragg's law for Bragg resonances of water waves excited by five types of artificial bars[J]. Physics of Fluids, 2024, 36(4): 047130. doi: 10.1063/5.0201300 [46] KAR P, SAHOO T, MEYLAN M H. Bragg scattering of long waves by an array of floating flexible plates in the presence of multiple submerged trenches[J]. Physics of Fluids, 2020, 32(9): 096603. doi: 10.1063/5.0017930 [47] LIU H W, XIE W J, XIE J J, et al. Zero reflections of water surface gravity waves induced by a finite periodic array of artificial bars[J]. Physics of Fluids, 2025, 37(3): 037163. doi: 10.1063/5.0255928 [48] MILES J W. Oblique surface-wave diffraction by a cylindrical obstacle[J]. Dynamics of Atmospheres and Oceans, 1981, 6(2): 121-123. doi: 10.1016/0377-0265(81)90019-1 [49] LIU H W, ZHOU X M. Explicit modified mild-slope equation for wave scattering by piecewise monotonic and piecewise smooth bathymetries[J]. Journal of Engineering Mathematics, 2014, 87(1): 29-45. doi: 10.1007/s10665-013-9661-6 [50] 滕斌, 候志莹. 变化地形上波浪传播模拟的二维BEM模型[C]//第十九届中国海洋(岸)工程学术讨论会论文集. 重庆, 2019: 379-384.TENG Bin, HOU Zhiying. Two dimensional BEM model for modelling of wave propagation over varying topographies[C]//Proceedings of the 19 th China Ocean (Shore) Engineering Symposium. Chongqing, 2019: 379-384. (in Chinese) -
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