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                     森羅万象セミナー  第 76 回
               http://www.ep.sci.hokudai.ac.jp/~sinra/
                               
                           Dr. M.M. Popov
                (V.A. Steklov Mathematical Institute, 
	 Russian Academy of Sciences St. Petersberg, Russia)


              『On some peculiarities of ray tracing 
                in inhomogeneous anisotropic media』

                 日時: 10 月 17 日 (木) 16:30 -- 17:30
                        場所: 5 号館 201 号室

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[Abstract]
Application of ray method in  inhomogeneous anisotropic media faces, in
addition to the caustic problems, some  specific problems when the phase
velocities of qS waves coincide. Mathematical  basement of the problems is
caused in particular by the fact that the  eigenvalues and eigenvectors of
the Christoffel tensor in this case loose, in  general, smoothness and that
gives rise to singularities in ray formulas. It  resembles, at this point,
well known caustic problems in ray theory and,  unfotunately, it turns out
to be rather complicated from Math point of view.

 To develop a physical  insight to the wave process in a
vicinity of the points where the velocities of  qS waves coincide we study,
from one hand, a simplified model problem but, from  the other hand, we
treat it in mathematically rigorous way. Therefore the  results are reliable
and do not contain any heuristic assumptions.
 The model can be interpreted as two  inhomogeneous elastic strings
interacting via a potential. The velocity of the  first string (channel 1)
and the second one (channel 2) coincide at a point x_*.
 The result can be described as follows.
Suppose one wave propagating with the  velocity of the first channel is
generated at
the point x=0 from the left side  of the point x_*. When the wave reaches
the point x_* a resonance phenomenon  takes place and on some distance
behind x_* we observe appearence of a new wave  which propagates with the
velocity of channel 2. Its amplitude, or the  diffraction coefficient,
germinates from the resonance domain near the point  x_*.
Based on this model problem, we can  anticipate that in the general case of
anisotropic inhomogeneous media i) the  ray ansatz fails in the resonance
domain and
should be replaced by some more  complicated formula, ii) starting with only
one incident qS wave before the  resonance domain we shall get both of them
behind it and the second wave may be  of the same order as the initial one.

 It is supposed in the presentation:
1.To give a popular intoduction to wave propagation problems in Geophysics,
2.To illuminate main ideas of the ray method and physical interpretation of
ray formulas,
3.To show advantagies and disadvantagies of ray method,
4.To point out mathematical basement of resonance problems in inhomogeneous
anisotropic
 media,
5.To show our main results for the above mentioned model problem omitting
mathematical
 details.

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北海道大学大学院 理学研究科 地球惑星科学専攻
惑星物理学研究室 博士課程 1 年

高山 歌織 (TAKAYAMA Kaoru)

E-mail: kaorun@ep.sci.hokudai.ac.jp
Tel: (011)706-4494 (内線 4494)