Symplectic Geometry Theory And Wavelet Transform Method In The Application Of High Frequency Approximation Of The Wave Equation

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.