PROPAGATION OF ELASTIC WAVES IN LAYERED MEDIA BY ...

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376 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA(INTERFACE)UPPERMEDIUMn=N-In=Nm-In=N+l -- 4- -I-tI, !-m-':' IIIII---1I. . . . III=-rn= I (INTERFACE)n=2n=3LOWERMEDIUM(a)UPPERMEDIUM*(INTERFACE)n:N-2 -- -n=N-In=Nm-III . . . . .Irn m+ln=l~r n---2---I-- - --IIIIIIIIIin=3LOWERMEDIUMZ(b)FzG. 5. Grid arrangement at the interface z = h. Part (a) shows the grid points used whenthe upper medium is extended to include an additional fictitious line below the interface, andpart (b) shows the grid points used when the lower medium is extended to include an additionalfictitious line above the interface.approximation. Equation (9) then becomes1 B 1 1 1 A 1 A 12~r [ .~+I,N,~ - Bm-I,N,~] + ~ [ m,~+l,~ -- m,~,~](#~)I1 2 B 2 1 A~ A2 1= ~ (B~+1,1,, - ~-~,1,,) + ~ ( ,~,~,,- ,~,~,,) • (25)
PROPAGATION OF ELASTIC WAVES IN LAYERED MEDIA 377The notation involving the second subscript on A and B is important. In medium 1,the subscripts N and N 4- 1 indicate the interface and fictitious line, respectively.In medium 2, the subscripts 1 and 2 indicate the interface and next line beyond theinterface extending into medium 2. From equations (10) and (11), which requirethe continuity of displacement across the interface, we have2 _- A 1Am,l,p m,N,pB m=t=1,1 ~ ,p = B m+l 1 ,N,p •(26)Substituting into equation (25), and solving for A ~ on the fictitious line, we obtainA' [ (::)]A 1 (~)A 2m,N+l,p = 1 -- m,N,p 4- m,2,p1 (~__z~-~rl _ (p,2)] 1 1- 2 \~r/L ~ [B,~+~.N,p-- Bm-~,~-,,]. (27)In a similar way, the other interface condition (equation (8)) gives1 p2 Yc2 B 1 p2 vc2 B 2B.,,N+,,. = E 1- Pl \~J I '~'~'" + m \Vcl--] m,2,p1 A(~r)([ (va~ 2] p¢~) r(v~2,)2 2 (v.2~2]~-~ 1-2 . . . .[ , i 2,]X A,~+l,~v,~, -- A~-LN,p + -A.,,N,p • (28)mAlong the axis of symmetry (m = 0) the values of A and B on the fictitious linebecomeA~,N+~,, = 0 (29)and1 [ p2 vc2 ;1 B~,N,p 4- 02 vc2 ; 2B o,N,I,p--- 1- ~ \Vd--/ _1 Pl \~cl/ B o,2,p(Az)([~ (Vsl~2]- P(~pl)r(vc2]2\~)Cl\vc:/ ._1(t;s2)2]} 1~cl-- 2 1 -- 2 -- Li-- I -- 2 AI,N,,.(30)The special considerations required here have been explained earlier.Note that the determination of A t and B 1 on the fictitious line N 4- 1 at time pmeans that we can compute A 1 and B ~ on the interface at the next time step p 4- 1by using the equations of motion for medium 1. Also, since we can compute A ~ andB ~ on the interface (denoted by n = N in medium 1), we know A 2 and B 2 on the
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