2002 Fiscal Year Final Research Report Summary
Analysis of scattering waves by perturbed portions
| Project/Area Number |
13640150
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | IBARAKI UNIVERSITY |
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Principal Investigator |
SOGA Hideo IBARAKI Univ. College of Education, Professor, 教育学部, 教授 (40125795)
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| Co-Investigator(Kenkyū-buntansha) |
KAWASHITA Mishio HlROSHIMA Univ., College of Science, Assoc. Professor, 理学研究科, 助教授 (80214633)
TANAKA Yasuo IBARAKI Univ., College of Education, Professor, 教育学部, 教授 (30007520)
KAIZU Satoshi IBARAKI Univ., College of Education, Professor, 教育学部, 教授 (80017409)
ITO Hiroya Univ. of Electro-Comm., Dept Math. Assoc. Professor, 電通学部, 助教授 (30211056)
NAKAMURA Gen HOKKAIDO Univ., College of Science, Professor, 理学研究科, 教授 (50118535)
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| Project Period (FY) |
2001 – 2002
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| Keywords | the wave equation / elastic equations / scattering theories / reflection of waves / partial differential equations / hyperbolic equations / energy decay / inverse problems |
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| Research Abstract |
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
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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
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