Analysis of special waves in elastic bodies

弹性体中的特殊波分析

• 批准号: 10640151
• 负责人: SOGA Hideo
• 金额: $ 1.86万
• 依托单位: Ibaraki University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640151/
• 关键词: Elastic Waves; Wave Equations; Scattering Theory; Asymptotic Solutions; Reflection; Energy Decay; Mathematical Physics; Hyperbolic Equations; 弾性方程式; 表面波; 全反射; 逆問題; 双曲系方程式

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.

本研究计划是关于弹性波的，最初的主要目的是：a）获得解的具体表示，B）研究波在边界附近的散射，c）研究关于边界附近波的反问题，d）分析全反射情况下的波，我们几乎按预期完成了这些工作。我们总结一下本项目的结果，对于a）和d），我们得到了全反射波的渐近展开式，这是最主要的结果之一。这一展开式在分析弹性波的散射现象和反问题中具有重要的应用价值。关于B），我们证明了瑞利波的独立性，并可以从Lax-Phillips散射理论中提取出来：我们给出了瑞利波的理论公式，这对反问题是有用的。此外，我们还精确地研究了这种波在无穷远处的衰减，对于c），我们得到了一种适用于反问题的重构微分方程系数的新方法。并给出了一种新的有限元数值方法。这些方法对求解反问题似乎很有用，但有些反问题我们还不能具体地解决。