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浅水波越过海底周期圆台阵列的能带结构

发布时间:2018-04-04 18:26

  本文选题:能带结构 切入点:Bloch原理 出处:《广西民族大学》2016年硕士论文


【摘要】:自上世纪90年代以来,在固体力学中具有广泛应用的能带理论被拓广到研究液体表面波在海底周期排列的无穷阵列结构上的传播并取得大量成果.然而很遗憾,二十多年来国际上所有的相关研究,几乎全都局限于简单的分段或分片常数深度的底部结构,如二维矩形台阶系列和三维(出水、水下或底部钻孔)直立圆柱阵列或直立棱柱阵列等,而周期性排列的变水深无限结构阵列几乎从未被研究过,原因在于变水深情形下,若采用浅水波方程为控制方程,则需要求解变系数偏微分方程或常微分方程,若采用叶氏方程或者缓坡类方程,由于线性波色散关系中波数为水深函数的隐函数,则需要求解系数为隐函数形式的偏微分方程或常微分方程.本学位论文研究线性浅水波越过海底周期排列的无限圆台阵列的能带结构.显然,在圆台之上的水深是连续变化的,称为变水深,这是以往有关液体表面波能带结构研究中极少涉及的.因为本文考虑的是线性浅水波,所以采用的控制方程为线性浅水波方程(也称线性长波方程).基于二元周期函数的傅里叶级数展开理论,我们将周期变化的水深函数展开成傅里叶级数,同时也将所求的自由液面高程函数的周期性部分展开成傅里叶级数,系数待定.然后将它们双双代入控制方程,并对傅里叶级数的无限求和进行有限截断,将原本是求能带结构的问题转化为求矩阵特征值的问题.最后,我们计算出了浅水波在按正方晶格和六角晶格两种方式排列的无穷周期阵列上的能带结构,并发现针对某些地形参数,正立圆台或倒立圆台都可能形成相应的完全频隙,所谓完全频隙是指在任何传播方向,频率落在此频隙区间的波在相关周期地形上都是绝对禁止或无法存在.进一步,我们分析讨论了不同地形参数尤其是圆台上下底面的填充率对频隙宽度以及频隙位置的影响.所得结果对近海工程中有限周期排列结构物的建造和优化具有理论性的参考价值.
[Abstract]:Since the 1990s, the energy band theory, which has been widely used in solid mechanics, has been extended to study the propagation of liquid surface waves on the periodic array structure of the seabed, and a lot of results have been obtained.Unfortunately, however, for more than two decades, almost all of the relevant studies have been confined to simple bottom structures with constant depths of segments or slices, such as two-dimensional rectangular step series and three-dimensional (effluent).Underwater or bottom drilling) vertical cylindrical arrays or vertical prism arrays, etc., but periodic arrays of infinite structures with variable water depth have almost never been studied. The reason is that in the case of varying water depth, the shallow water wave equation is used as the governing equation.It is necessary to solve the partial differential equation or ordinary differential equation with variable coefficients. If the Yehlet equation or the gentle slope equation is used, the wave number in the linear wave dispersion relation is an implicit function of the water depth function.Then it is necessary to solve partial differential equations or ordinary differential equations with implicit coefficients.In this dissertation, we study the band structure of linear shallow water waves across an infinite array of circular arrays arranged periodically on the seafloor.It is obvious that the water depth above the platform is continuously varying, which is rarely involved in the previous researches on the energy band structure of liquid surface waves.Because the linear shallow water wave is considered in this paper, the governing equation is linear shallow water wave equation (also called linear long wave equation).Based on the Fourier series expansion theory of binary periodic function, we expand the periodically varying water depth function into Fourier series, and at the same time, we expand the periodic part of the free liquid level elevation function into Fourier series, and the coefficients are to be determined.Then they are replaced into the governing equation and the infinite sum of Fourier series is truncated finitely. The problem of finding energy band structure is transformed into the problem of finding the eigenvalue of matrix.Finally, we calculate the energy band structure of shallow water waves on infinite periodic arrays arranged in square lattice and hexagonal lattice, and find that for some terrain parameters, both orthotropic and inverted stations may form a corresponding complete frequency gap.The so-called complete frequency gap means that the wave falling in the frequency interval in any direction of propagation is absolutely forbidden or unable to exist on the related periodic terrain.Furthermore, the influence of different topographic parameters, especially the filling ratio of the bottom surface of the platform, on the frequency gap width and the frequency slot position is analyzed and discussed.The obtained results are of theoretical reference value for the construction and optimization of finite periodic structures in offshore engineering.
【学位授予单位】:广西民族大学
【学位级别】:硕士
【学位授予年份】:2016
【分类号】:P751;O481.1

【参考文献】

相关期刊论文 前3条

1 钟兰花;吴福根;钟会林;;Effects of orientation and shape of holes on the band gaps in water waves over periodically drilled bottoms[J];Chinese Physics B;2010年02期

2 钟兰花;吴福根;;水波在周期性钻孔底部结构中的传播及其能带[J];物理学报;2009年09期

3 许泰文,张宪国,蔡立宏;Bragg Reflection of Waves by Different Shapes of Artificial Bars[J];China Ocean Engineering;2002年03期



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