缓坡类方程的解析解与逼近解析解及港口防波堤优化设计的基础研究

缓坡类方程因其高效的计算能力（因为维数比Laplace方程少一维）和广泛的波谱适应性（涵盖长中短波）以及包含了折射衍射双重效应等诸多优点而成为线性波领域应用非常广泛的方程。但由于色散方程的隐含关系导致缓坡类方程的系数也是隐含的，极大地增加了求理论解析解的难度。在国家自然科学基金前期资助下，缓坡类方程的解析解研究已经取得大的突破，隐式修正缓坡方程在水深函数为分段单调函数的情形下已被成功转换为显式修正缓坡方程，且针对三个理想海底地形，已经构造了修正缓坡方程的准确解析解。本项目拟在我们前期突破性工作基础上，进一步给出修正缓坡方程相对于各种拟理想变水深海底地形的准确解析解和具弱非线性效应修正缓坡方程的逼近解析解，同时对布拉格防波堤的共振反射和港口振荡两大类共振现象进行解析模拟，利用多元函数的极值理论，对布拉格防波堤和港口基础设计的优化进行理论探讨。

Mild-slope type equations (MS type equations) are most powerful equations in linear wave regime due to their highly efficient computing ability, broad validity in wave spectrum and inclusion of both refraction and diffraction effects. However, it is extremely difficult to construct analytic solutions to the MS type equations since their coefficients are implicit due to the implicit linear dispersion relation. Thanks for the prophase support from the NSFC, we have made an important breakthrough in seeking analytic solutions to the MS type equations, and the implicit modified mild-slope equation (MMSE) has been successfully transformed into explicit form. Based on the explicit MMSE, three analytic solutions have been constructed for idealized topographies. In this project, we are going to construct a series of analytic solutions to the MS type equations for quasi-idealized topographies with variable water depth and approximate analytic solutions to weakly nonlinear MS type equations. In addition, we are going to conduct analytic modelling to Bragg resonant reflection and harbor oscillation. By using extreme value theory, we are going to conduct intensive investigation to optimal design of Bragg breakwaters and harbors.