Analysis of special waves in elastic bodies

本研究项目涉及弹性波，最初设定的主要目的是：a)得到弹性波解的具体表示，b)研究边界附近波的散射，c)研究边界附近波的逆问题，d)分析全反射情况下的波。我们几乎按照预期完成了这些工作。让我们总结一下这个项目取得的成果。关于a)和d)，我们得到了全反射波的渐近展开式，这是最重要的结果之一。这种展开有望在分析弹性波的散射现象和反问题方面有所帮助。此外，我们还得到了波浪的另一种具体表示形式。关于b)，我们已经证明了很久以前就非常感兴趣的瑞利波具有单独的行为，并且可以在拉克斯-菲利普斯散射理论中提取出来。我们已经为瑞利波提出了该理论的公式，这对逆问题很有用。此外，我们还精确地研究了这种波在无穷远处的衰减。关于c)，我们得到了一种适用于反问题的微分方程系数重构的新方法。得到了近似解的一种新的有限元数值求解方法。这些似乎对解反问题很有用，但是我们不能完成一些反问题的具体解。

This research project is concerned with elastic waves, and the main purposes set initially werea) to obtain concrete representations of the solutions,b) to study scattering of the waves near boundaries,c) to study inverse problems concerning the waves near boundaries,d) to analyze the waves in the case of the total reflection.We have accomplished these almost as was expected. Let us summarize the results obtained in this project.About a) and d) , we have got an asymptotic expansion of the wave reflected totally, which is one of the mainest results. This expansion is much expected to be useful in analyzing the phenomenon of the scattering and the inverse problems of the elastic waves. Furthermore, we have obtained also another kind of concrete representation of the waves.About b) , we have shown that the Rayleigh wave, which has been much interested since a long time ago, behaves individually and can be extracted in the Lax-Phillips scattering theory : We have made a formulation of that theory for the Rayleigh wave, which is useful for the inverse problems. Moreover, we have investigated precisely decay of this wave expressing the behavior at infinity.About c) , we have got a new methods of reconstruction of coefficients in differential equations applicable to the inverse problems. And we have obtained a new numerical method of finite elements for the approximate solutions. These seem to be very useful to solve the inverse problems, but we cannot finish solving concretely some of the inverse problems.