PROPAGATION OF ELASTIC WAVES IN LAYERED MEDIA BY ...

B~-I,~,~]378 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICAinterface (denoted by n = I in medium 2) by continuity of displacements (equations(10) and (11)). Hence we can find A 2 and B 2 at the next time step p -P i using theequations of motion for medium 2.Equations (27), (28) and (30) contain the ratios (#2/t~1) and (v~2/v~l)2(p2/pl). Ifthe constants characterizing the two media are quite different, then these ratiosmay be large. The effect of these large ratios is to introduce serious errors and(sometimes) instabilities in the calculations. The reaso~ this occurs is because theone-sided derivative in z introduces an asymmetry in the boundary conditions. Ifwe had used a two-sided or centered derivative in z, the finite difference equationsfor the boundary conditions would be symmetrical. However, the calculation schemeacross the interface would no longer be explicit. When the ratios (t~2/pl) and(v~2/v~) 2(p2/pl) are large, we add the fictitious line to medium 2 (rather than medium1) and then derive lesults for A 2 and B 2 along this fictitious line. See Figure 5b.Proceeding in the same manner as before, we obtain the following results whenm~0:= -A~,l,p 2 # 1(Art) (t~) 1 -- [Bm+l,N,p, i+ ( 1 -- m,.~,p-FP,I 1- Am,N-l,p (31)tt2B 2 pl v~ B 1 pl v~l B 1-- m,N--l,pm,l,p = 1- P22 \f)c2/ I m,N,p + p22 \Vc2/Az 1()\Vc2/ J p2 L\Yc2/X[A 1 A t 2A1 1m-~-l,N,p -- m--l,N,p ~ -- m,N,p •m(32)Along the axis of symmetry (m = 0) we haveI (2( (2(2 pl v~l i Pl v~l 1Boa,. = 1 -- B0,N,p + -- B0,~-l,.\vc2/ j \v~2/A ~ 0,1,p = 0 (33)\v~2/ I ~ L \V~2/ \v~21 j)In comparing equations (27)-(30) with equations (31)-(34), we see that the effectof changing the fictitious line from medium 1 to medium 2 is to invert the ratios(p~/~l) and (vc2/v~l)2(p2/;l) into (~1/~2) and (Vcl/Vc2)2(pl/p2), respectively. In ourcomputational work we use the set of equations for which the ratios are least.(4) Special Treatment of Source. The point compressional source used in our work