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U_(N-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U_(N-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.
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,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.
作者
谢文洁
谢健健
刘焕文
XIE Wenjie;XIE Jianjian;LIU Huanwen(School of Marine Engineering Equipment,Zhejiang Ocean University,Zhoushan,Zhejiang 316022,P.R.China;School of Hydraulic and Civil Engineering,Ludong University,Yantai,Shandong 264025,P.R.China)
出处
《应用数学和力学》
北大核心
2025年第9期1176-1195,共20页
Applied Mathematics and Mechanics
基金
国家自然科学基金(51879237)
浙江省自然科学基金(LQN25E090005)。
