稳态的布拉格精确共振和近似共振水波研究

Traditionally, the perturbation and multiscale method are used to study the time evolution of the wave amplitude when the exact and near Bragg resonance occurs in water waves. Usually, the wave amplitude of the resonant component is assumed to be zero initially. ..From the viewpoint of fully-developed seas where the reflected wave component has non-zero wave amplitude initially, the proposed project will investigate analytically and numerically three classes of both exact and near Bragg resonance when the fully-developed seas propagate over 2D and 3D undulated bottoms. The exactly nonlinear wave equations will be solved by the Homotopy Analysis Method. The Zakharov equation for the undulated bottom will be derived and solved directly by the software of Mathematica. Multiple steady-state resonant wave systems which have time-independent wave amplitudes are supposed to be obtained. The effects of mean water depth, incident angle and slope, bottom slope and the extent of the near resonance on the steady-state wave systems will be studied in details. The corresponding bifurcations according to various physical parameters will hence be found...This project can deepen our understanding on the exact and near Bragg resonance in water waves. It has great scientific value and means for the fundamental research.

水波理论中的布拉格共振研究，传统上通常采用摄动理论或多重尺度等方法，在假定共振反射波波幅初始状态为零的视角上，研究共振波波幅随时间的演化规律。..本项目拟从共振反射波波幅不为零的充分发展波浪之研究视角出发，运用同伦分析方法求解精确的非线性波浪方程，推导适用于非平坦水底的Zakharov方程并利用符号计算软件直接数值求解该方程，从解析和数值两方面研究充分发展波浪在二维和三维水底上传播时发生的三类布拉格精确共振和近似共振问题，获得共振时波幅不随时间变化的稳态波系，并考察其多样性，分析该稳态波系受到平均水深、入射角度、入射波波陡、水底起伏大小以及近似共振的近似程度等物理因素的影响而发生的分叉现象。..该项目的研究有助于大大加深我们对水波中布拉格精确共振和近似共振现象的理解和认识，具有重要的理论基础研究价值和科学意义。

水波理论中的共振问题研究，传统上通常采用摄动理论或多重尺度等方法，研究共振波波幅的演化规律，即传统上认为该波幅从零开始随入射波在水底传播距离的增加，首先线性增长、而后周期变化的规律。本项目基于同伦分析方法这一解析方法，将其在处理波浪共振方面进行了优化，为更全面的研究浅水下的布拉格共振问题提供强有力的支撑。重点讨论了有限振幅的共振波浪系统，同时考察近似共振对整个波浪系统所能够产生的影响。对不同水深情况下具有有限振幅的共振波浪系统进行了系统地研究，成功地获得了该问题中波浪系统的稳态解。更为重要的是，针对近似共振的研究，进一步揭示了在共振波浪系统中，精确共振和近似共振之间的联系与区别。