弹性波在复合材料中的传播问题是新材料领域发展的重要方向之一, 而对波 动方程Green 函数的研究是一项非常困难的工作. 2015 年, 国际数学家大会的45分 钟报告人、新加坡国立大学的尤释贤( S.H. Yu) 教授、王维克教授和邓师瑾老师得 到了上半空间的Green 函数[3]. 而有界区域上分片常系数波动方程的Green 函数 目前尚无结果. 2016 年, 董弘桀教授和李海刚老师得到了内含物是两个距离为?? 的圆形区域 上散度型椭圆方程传导问题的梯度估计, 其中的主要工具就是构造分片常系数椭 圆方程的Green 函数[2]. 为研究弹性波在复合材料中的传播行为, 我们也想构造分 片常系数波动方程的Green 函数. 本学位论文利用分离变量法在构造一维有界区间上分段常系数波动方程的 Green 函数方面取得了进展. 本文建立了系数为分段常数1和??2(?? ∈ N+) 的波动方 程Green 函数的正交三角级数表达式, 并突破性地确定了每一项的系数, 其中一部 分项的系数由通项公式确定, 其余项的系数可由已知项的系数迭代得到, 这为研究 弹性波在复合材料中的传播问题提供了有力的工具. 此外, 本文还进行了其它的探索: (??) 利用像源法得到了上半空间的Green 函 数; (??) 利用多像源法和Laplace 变换法得到了一维有界区间上Green 函数的级数 表示, 说明了波动信号在边界处被适当反射; (??) R.E. Kleinman [8] 给出了当波动数 充分小时, Helmholtz 方程在外区域的Green 函数, 本文补充证明了这一重要结果; (??) 给出了L.C. Evans [13] 提出的双曲反演定理的详细证明过程.

The propagation of elastic wave in composite materials is one of the most important directions in the development of new materials, the study of Green functions for wave equations is a tough project. In 2015, S.J. D、professor W.K. Wang, and professor S.H. Yu, at NTU of Singapore, who delivered a 45-minute report at ICM, obtained the Green functions in R?? + × Rd+. As for wave equations with piecewise constant coefficient in bounded domain, there is no progress until now. In 2016, H.J. Dong and H.G. Li established the gradient estimates for elliptic equations in divergence form conductivity problem, where the domain contains two disks with a distance of ??. Their main idea is to construct the Green functions for elliptic equations with piecewise constant coefficient. In order to study propagation behavior of elastic wave in composite materials, we also try to construct the Green functions for wave equations with piecewise constant coefficient. In the thesis, we achieve breakthroughs in constructing the Green functions with piecewise constant coefficient 1 and ??2 (?? ∈ N+) in bounded interval by using separation of variables. We establish the Green functions in sine series form, where the coefficients of some terms are determined by the formula, and other coefficients can be obtained by iteration of the coefficients of known terms. The result provides a useful tool for studying elastic wave propagation in composite materials. In addition, we also make some other attempts as follows: (??) We obtain the Green functions in half upper space by using image method; (??) We get Green functions in bounded interval by multiple images method and Laplace transform method, which show that the signal is reflected on the boundary; (??) When ?? = 3, R.E. Kleinman [8] solved the Green functions for Helmholtz equations in exterior domain for sufficiently small wave number, we supplements the proof in more detail;(??) We prove the hyperbolic Kelvin transform theorem(see in [13]) in very detail.