辛几何理论和小波变换方法在波动方程高频近似中的应用

【作者】 陈东方；

【导师】 韦穗； 李世雄；

【作者基本信息】 安徽大学 ， 计算机应用， 2003， 博士

【摘要】 本论文探讨了Maslov方法在波动方程高频近似中的应用及焦散区的具体求解方法；讨论了用小波变换化简二维非均匀缓变介质条件下的波动方程；本文还讨论了凹面反射的焦散现象、凹面反射焦散区的奇性种类，讨论了二、三维凹面反射波动场非焦散区、焦散区的计算方法。本论文的工作作为国家自然科学基金地震波传播与成像项目(NO. 40174032)的一部分，得到如下结果： (一)论文分析了利用Maslov方法求解非均匀介质中波动方程高频近似解的基本原理和求解过程，构造了介质系数只在一个方向变化时波动方程高频近似解的通用计算公式。通过引入波向量(慢度向量)，将物理空间中几何光学的射线问题转化为辛空间中的Lagrange子流形(超曲面)问题。由于出现焦散现象的原因在于Lagrange子流形在该处的切平面与物理空间垂直，通过转换适当的投影方向，然后将这个投影方向上得到的高频近似解再变换回到原来的物理空间中，得到了在焦散附近适用的高频近似解。文中还给出了计算实例。 (二)构造了利用小波变换简化二维非均匀缓变介质中波动方程的近似方法。利用小波变换在空间域和频率域上具有局部性的特点，通过小波展开，把二维非均匀问题转化为一系列一维非均匀问题，然后利用在(一)中得到的通用计算公式求解，大大降低了问题的复杂性和难度。 (三)深入讨论了凹面反射波动场的焦散现象、焦散区的几何结构与奇性的特点以及非焦散区、焦散区波动场的计算问题。论文主要包括三个方面：(1)分析了凹面反射的焦散现象，给出了不同凹面反射的焦散图；(2)分析了二维凹面反射波动场焦散现象产生的原因及焦散区奇性的种类，得出了二维凹面反射波动场焦散区奇性主要有折叠(fold)和尖点(cusp)两种的结论，利用辛几何方法构造了圆锥曲面反射波动场非焦散区和焦散区的通用计算公式，并给出了圆柱面、椭圆柱面及双曲柱面反射的计算结果；(3)分析了三维凹面反射波动场焦散现象产生的原因及焦散区奇性的种类，得出了三维凹面反射波动场焦散区奇性主要有折叠（foM、尖点kusP和燕尾k）三种的结论，提出了利用辛几何方法计算三维凹面反射波动场非焦散区和焦散区的计算方法，并给出了三轴不等椭球体凹面反射波动场的计算结果剖面图。 论文还就今后所要开展的工作进行了分析和讨论。更多还原

【Abstract】 This thesis discusses Maslov’s method, wavelet transform and their applications to asymptotic evaluation of wave equations in high frequency fields. The method of solving wave equations in caustic domain by symplectic geometrical theory, and the method of simplifying two-dimension wave equations in slowly varying nonhomogeneous medium by wavelet-transform theory are proposed. The caustics phenomena of electromagnetic wave propagation in concave reflector, the singularities of caustics, and the method of computing wave fields in and far away from the caustics in concave reflector are also discussed. The important parts of this work consist of:1. The high frequency asymptotic evaluation of wave equations in nonhomogeneous medium by Maslov’s method is systematically studied, and formulations of the high frequency asymptotic evaluation in nonhomogeneous medium which varying only in one direction are constructed. While the new components having the same numbers with these original physical vectors are introduced and the new components are combined with those original physical components to form a new symplectic space, the ray problem of wave propagation in geometrical optics is converted into the problem of Lagrange submanifold in the symplectic space. Since the cause of caustics phenomena is that the tangent plane of Lagrange submanifold in caustic fields is perpendicular to the original physical space, we solve the high frequency asymptotic problem in a new mixed space by changing the projecting direction, then we get the high frequency asymptotic solutions of wave equations efficiently near and on the caustics.2. The approximate method of simplifying two-dimension wave equations in slowly varying nonhomogeneous medium is constructed. Being local in space and frequency, and some even compactly supported, wavelets are used to simplify the wave equations in slowly varying nonhomogeneous medium, and transfer the problem of solving two-dimension wave equations into a series of one-dimension problems, after words, the method in 1 can be applied to solve it. As a result, the complexity ofthe problem and the difficulty in solving it are largely reduced.3. The caustic phenomena of electromagnetic wave propagation in concave reflector, the singularities of caustics, and the method of computing wave fields in concave reflector are discussed in detail. The following three parts are included: (1) The caustic phenomena of electromagnetic wave propagation in concave reflector is studied, and the pictures of caustic fields in different concave reflectors are displayed. (2) The causation that the caustic phenomena of electromagnetic wave propagation in two-dimension concave reflectors occurs and the types of singularities in caustic fields are investigated, and the conclusion that there are two types of singularities (fold and cusp) in caustic fields in two-dimension concave reflectors is obtained; By symplectic geometrical method, formulations of computing wave fields in and far away from caustic fields in two-dimension concave reflectors are deduced, and the results are plotted in pictures. (3) The cause of the caustic phenomena of electromagnetic wave propagation in three-dimension concave reflectors and the types of singularities in caustic fields is discussed, and the conclusion that there are three main types of singularities (fold, cusp and swallowtail) in caustic fields in three-dimension concave reflectors is obtained; By symplectic geometrical method, the formulae of computing wave fields in and far away from caustic fields in three-dimension concave reflectors are deduced. Particularly, the wave fields in ellipsoid concave reflector are computed, and the results displayed in special sections are given.The further works about this topic are also addressed.更多还原