Analysis of scattering waves by perturbed portions

扰动部分的散射波分析

基本信息

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

项目摘要

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

本研究项目涉及具有初始扰动部分的介质中的弹性波，其主要目的如下：（a）对各种散射理论进行分类，并提取它们的特征点，并为这些理论制定新的理论。 (b)利用(a)的结果来研究具体的散射问题，特别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。我们有这些几乎达到了预期。让我们总结一下这个项目中获得的结果。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，它们被视为结论性理论之一。主要结果是我们已经阐明了这些类型之间的联系，提取了它们的特征，即证明了y可以通过一定的过程进行交换，并且利用这个结果，我们构建了一个散射理论。 Lax-Phillips 类型适合于弹性表面波的研究，并且似乎成为这些波的散射反演问题的基础作为另一个主要结果，我们已经在一般数学框架上表明，在弹性表面波中必定会出现特殊类型的波。关于（b）：我们得到了由弹性介质中的孔反射的波的渐近展开，这是非常具体的（不是工程意义上的）。他们可以申请它可以用于反演问题，例如，从反射波的数据得知孔中是否充满液体。我们将扩展扩展到不连续波被完全反射的情况。为了证明这一点，结果为（a。）也被使用，我们还精确地检查了瑞利波（表面编织之一）的衰减，这意味着该波在能量意义上集中在表面上。