Ishii  Hiroshi

Robert  M.Ellis

English

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Ishii, Hiroshi

Ellis, Robert M.

Tohoku University Repository (TOUR) / 東北大学機関リポジトリ

Expression of a Free Rayleigh Wave UsingComplex Angles著者 Ishii  Hiroshi, Robert  M.Ellis雑誌名 Science reports of the Tohoku University. Ser.5, Geophysics巻 20号 1ページ 45-47発行年 1970-07URL http://hdl.handle.net/10097/44699 The Science Reports of The  TOhoku  University, Ser. 5,  Geophysics,  Vol. 20, No. 1, 1970Expression of a Free Rayleigh Wave Using Complex Angles  HIROSHI  ISHII            Geopyhsical Observatory, Tohoku University, Akita, Japan                             and  ROBERT M. ELLIS Department of  Geophysics, University of British Columbia, Vancouver B.C., Canada  Received July 1, 1970) Abstract: Complex angles were successfully used for the calculation of     amplitude characteristics for incident plane P and  Sr' waves in the case of a dipping  laver by Ishii and Ellis (1970b).  Tnthis paper, it is shown that a free Rayleigh wave       is also expressed using complex angles. 1.  Introduction  Complex angles were used in order that the cases of  total reflection and incident angles greater than the critical angles were involved in the results in the calculation of the  displacements in the case where a dipping layer exists (Ishii and Ellis, 1970h).    The expression f Rayleigh waves in terms of complex angles is also of interest for further studies of surface waves. Therefore, the present study will give an interpreta-tion  of free Rayleigh waves in terms of complex angles. 2. Formulation and Discussion    Let us consider an elastic half-space with free surface  0=0 (Figure 1). For plane waves propagating in the  x-y plane, the motion is independent of z and the displacement has only  r and  0 components. The dilatation 0,and rotation  CCI satisfy  p2(9  +k  0  ,                                           (1)                    p2co- lqw --- 0 , where  ka =  &VC,  ,  kb  --rolCb and Ca and  C  b are the P and S wave velocities respectively  (Ishii and Ellis,  1970b). The stress components are expressed by                   Pl2 a  7  1 aca   00  =pC,2ie +2CF,                   uar2ar r  o  /1}                                                (2)           r02pC2,, I a(1 ae 1 ata               •ukFT r  ao k2  ar2  ' where p is density. The solution in the medium can be written as46 H. ISHII and  R.M.  Free  surface ELLIS Ray  I  e  igh           Fig. 1. Coordinate system used to calculate the complex angle of  a  free  Rayleigh wave.                                    OR=AR'eikajc°s_aR)                                    (3)                                        =BR.eikbrcos (6-0R) where AR and BR are constants, and aR and  i3R are propagation directions measured from  0=0 as defined in the previous papers (Ishii and Ellis, 1970a nd 1970b).    The boundary conditions at 0=0 are  00  —  0  ,                                                (4) =  0  .Substituting (3) into the boundary conditions (4) using (2), we have  (1  —  2vg cos2  aR)A  R  471B  Rsin  /3R  cos  13R  =0  ,                                             (5) —  AR  sin aR  cos  aR +  (1-2 cos2 PR) BR =  0  ,and  vb  cos aR =  cos  PR  , (6) where  Pb =  CbIC. From  (5), we have  (1-24  cos2  aR)  (1-2 cos2 PR) —  4vg sin aR cos  aR  sin PR  cos  13R  —  0  . (7) Substituting (6), and writing  x=---cos2  aR and v  v; gives  16(1—v)x3  ±  (16—24x-„+8 x——1                             V3-=0 . (8) v  / v2 Assuming Poisson's relation,  X="2, yields  32  -  x3  —  56x2  +  72x  27  =  0  . (9)  3 The real rcot of this cquation is x=3.549 which corresponds to cos  aR=±1.884 and  using  (6)  , cos  PR  = ± 1.088. Recalling the relations      EXPRESSION OF A FREE  RAYLEIGH WAVE USING COMPLEX ANGLES 47                     arccos (—z) =  Tr — arccos  z,                 arccos  p =  i arccosh  p (fi=real>1) , (10) we obtain  aR  1.2471 or 7t —  1.2471  ,                                              (11) i8R =  0.4068i or  Tt  —  0.4068i  . If in equations (3), we use  cos (p±iq)  =  cos  p cosh q  i  sin  I) sinh  q  , (12)                 sin  (p±iq) =  sin  p cosh q  i  cos  p sinh  q  ,we have  eR  AR.  eAkar  (1  .881cos  6+11.597sine  )                                                          0.9218                                                               x coy                 AR  •  eiz 0.9194C1  C1  ( 13)  WR— BR  ,e=ikbr(1.088cose+i0.4278sin0)  x0.4278w                                = BR 'e±i 0.9194C1  CbY . We see that the dilatation and rotation propagate with the velocity 0.9194  Cb which coincides with the velocity of the free Rayleigh wave. As a result we see that for a Rayleigh wave written in terms of complex angles, the real part of the angle indicates the propagation direction and the imaginary part gives the decrease of amplitude with the two solutions of (11) representing waves propagating in opposite directions (0 and  7r).    This study was performed while one of the authors  (H.I.) stayed in the University of British Columbia.  Acknowledgments: This research was supported by the National Research Council (Grant  A-2617) and the Defence Research Board of Canada (Grant  9511-76). One of us  (H.I.) has been partially supported during this investigation by a University of British Columbia Graduate Fellowship.                                REFERENCES  Ishii, H. and R.M. Ellis, 1970a: Multiple reflection of plane SH waves by a dipping layer,          Bull. Seis. Soc. Am., 60, 15-28.  Ishii, H. and R.M. Ellis,  1970b: Multiple reflection of  plane  P and SV waves by a dipping          layer, Geophys. J.  R.  Ash'. Soc., (in press).

Expression of a Free Rayleigh Wave Using Complex Angles

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Expression of a Free Rayleigh Wave Using Complex Angles

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