The Zero Reflections Distribution of Linear Water Waves Crossing a Finite Periodic Array of Quasi-Idealized Artificial Bars

XIE Wenjie; XIE Jianjian; LIU Huanwen

doi:10.21656/1000-0887.450148

Volume 46 Issue 9

Sep. 2025

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Article Navigation > Applied Mathematics and Mechanics > 2025 > 46(9): 1176-1195

XIE Wenjie, XIE Jianjian, LIU Huanwen. The Zero Reflections Distribution of Linear Water Waves Crossing a Finite Periodic Array of Quasi-Idealized Artificial Bars[J]. Applied Mathematics and Mechanics, 2025, 46(9): 1176-1195. doi: 10.21656/1000-0887.450148

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The Zero Reflections Distribution of Linear Water Waves Crossing a Finite Periodic Array of Quasi-Idealized Artificial Bars

doi: 10.21656/1000-0887.450148

1. School of Marine Engineering Equipment, Zhejiang Ocean University, Zhoushan, Zhejiang 316022, P.R.China;
2. School of Hydraulic and Civil Engineering, Ludong University, Yantai, Shandong 264025, P.R.China

Funds:

The National Science Foundation of China(51879237)

Received Date: 2024-05-20
Rev Recd Date: 2025-06-12

Available Online: 2025-10-17

Abstract

Abstract

The zero reflection of phenomenon occurring with 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 U_{N-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 U_N-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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