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

本课题研究的是具有摄动部分的介质中的弹性波，最初设定的主要目的是：(a)对各种散射理论进行分类，并对每一类散射理论的特征点进行提取，并对研究不够充分的情况提出新的理论。(b)利用(a)的结果研究混凝土的散射问题，特别是关注工程中的逆散射问题，从散射波的数据中获取介质的信息。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)：波动方程的散射理论分为两种类型，即Lax-Phillips型和Wilcox型，它们被视为结论理论。其中一个主要的结果是我们澄清了这些类型之间的联系，提取了它们的特征。也就是说，通过一定的程序可以证明…同时，利用这一结果，我们建立了一个拉克斯-菲利普斯型的散射理论，该理论适合于弹性表面波的研究，似乎成为弹性表面波散射反问题的基础。作为另一个主要结果，我们在一般的数学框架上证明了在全反射情况下必然存在特殊类型的波，并进一步得到了这些波的渐近形式。关于(b)：我们得到了弹性介质中孔反射波的渐近展开式。这是非常具体的（不是在工程意义上），他们可以将其应用于反问题，例如，从反射波的数据中知道孔是否被液体填充。我们把这个展开式推广到不连续波完全反射的情况，用(a)中的结果来证明。我们还精确地研究了瑞利波（表面波的一种）的衰减，这意味着这个波在能量意义上集中在表面上。少

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）。