Analysis of scattering waves by perturbed portions

扰动部分的散射波分析

• 批准号: 13640150
• 负责人: SOGA Hideo
• 金额: $ 1.73万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640150/
• 关键词: the wave equation; elastic equations; scattering theories; reflection of waves; partial differential equations; hyperbolic equations; energy decay; inverse problems; 弾性波; 漸近解; 反射; 数理物理学

This research project is concerned with elastic waves in media with perturbed portions, and the main purposes set initially were as follow :(a) to classify various scattering theories together with extracting characteristic points for those groups of them, and to formulate new theories for the situations examined not sufficiently.(b) to investigate concrete scattering problems by means of the results in (a), especially, focusing our attention to inverse scattering problems in the engineering to get information of the media from data of the scattering waves.We have accomplished these almost as was expected. Let us summarize the results obtained in this project.About (a) : Scattering theories for the wave equations are classified into two types, that is, the Lax-Phillips type and the Wilcox one, which were treated as concluded theories. One of the main results is that we have clarified the connection between these types extracting their characteristics. Namely it has been proven that the … More y are exchangeable by a certain procedure. And also, using this result, we have constructed a scattering theory of the Lax-Phillips type which suites examination of the elastic surface waves and seems to become a basis for the scattering inverse problems of those waves. As another main result, we have shown on a general mathematical framework that there must appear special kinds of waves in the case of the total reflection, and furthermore have obtained asymptotic forms of those waves.About (b) : We have got an asymptotic expansion of the wave reflected by a hole in the elastic media. This is so concrete (not in the engineering sense) that they can apply it to inverse problems, for an example, to know from data of the reflected wave whether or not the hole is filled with a liquid. We have extended the expansion to the case that discontinuous waves are reflected totally For this proof the result in (a) is used. And also we have examined decay of the Rayleigh wave (one of the surface weves) precisely which means that this wave concentrated on the surface in the energy sense. Less

该研究项目与媒体中有扰动部分的弹性波有关，最初设定的主要目的如下：（a）对各种散射理论进行分类，并为其中那些群体提取特征点，并为所检查的情况提出新的理论，而不是充分检查的情况。散射波的数据几乎按预期完成了。让我们总结该项目中获得的结果。（a）：波动方程的散射理论分为两种类型，即Lax-Phillips类型和Wilcox One，被视为结论理论。主要结果之一是，我们已经证明了这些类型提取其特征的连接。也就是说，已经证明，……更多的y可以通过某个程序来交换。而且，使用此结果，我们已经构建了一种宽松的菲利普斯类型的散射理论，该理论套上弹性表面波的检查，似乎成为这些波的散射反问题的基础。作为另一个主要结果，我们在一般数学框架上表明，在全部反射的情况下，必须出现特殊的波浪，此外，这些波的渐近形式（b）：我们已经在弹性媒体中被孔反射的波反射不对称。这是如此具体（不是从工程意义上讲），以至于可以将其应用于反问题，例如，从反射波的数据中知道该孔是否充满了液体。我们已经将扩展扩展到了不连续的波完全反映的情况下，为此证明了（a）中的结果。而且，我们还检查了雷利波（表面韦斯的一个）的衰变，这意味着该波在能量上集中在表面上。较少的

项目成果

H. Soga: "Construction of asymptotic solutions of the elastic equations and their application"Theoretical and Appl. Mech. Japan 51. 309-314 (2002)

H. Soga：“弹性方程渐近解的构造及其应用”理论与应用。

川下美潮, 川下和日子, 曽我日出夫: "Relation between scattering theories of the Wilcox and-Phillips types and a concrete construction of the translation representation"Communication in Partial Diff. Equns.. (発行予定).

Mishio Kawashita、Wakako Kawashita、Hideo Soga：“Wilcox 和 Phillips 类型的散射理论与翻译表示的具体构造之间的关系”偏微分方程中的通信..（待出版）。