PROPAGATION OF ELASTIC WAVES IN LAYERED MEDIA BY ...

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(19)374 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICAIf we make these new substitutions in the equation of motion for B, thenBo,n,p-}-i ~- 2Bo,n,p -- Bo,n,p-1 .Jr_ k__~.._r ] k~z} [/~0,n+l,p --ZBo,n,p ~- Bo,n-l,p]+ 2 \ Ar ] \Az/L v~ [A~,,,+~,p -- A~,,,-,,~]+ 2 -Bo,.,.l.Equations (15)-(17) and (19) are the basic formulas used in our computations.We have carried out a standard stability study and find an approximate stabilitycriterion is given byAr /
PROPAGATION OF ELASTIC WAVES IN LAYERED MEDIA 375(22) are for n = --1 and therefore correspond to a fictitious line above the actualphysical surface at n = 0. See Figure 4.A complication arises on the axis of symmetry (m = 0) and special treatment isrequired. Due to continuity and cylindrical symmetry,A 1 0,-1,p = 0 (23)as in equation (17).An expression for B 10,-1,p can be found by using equation (6) for r,~ = 0 and replacingthe troublesome term (1/r)A ~ by OA1/Or in the same way as was done for theImTIim m+l_-: rn=-I (FICTITIOUS LINE)n=O (FREE BOUNDARY)n=lDELASTICMEDIUMZFIG. 4. Grid arrangement at the surface z = 0 showing the fictitious line and free boundary.equations of motion. Then we have\Ar/L \~/ _1 Al,o,p(24)where we use the symmetry condition A -1,~,p 1 = -Al,~,p. t Equations (21)-(24)make it possible at any given time level to compute values of A and B on thefictitious line n = --1. Knowing values on this line, it is then possible to use theequations of motion (15)-(17) and (19) up to and on the actual physical surface~ ~ 0.(3) Interface Boundary Conditions. The boundary conditions that must be satisfiedat the interface z = h are given by equations (8)-(11). In order to obtain anexplicit finite difference computational scheme, we first add a fictitious line tomedium 1 by extending the z-grid beyond the interface at z = h. A fictitious line isnot introduced in medium 2. The media have the interface z = h in common. Inmedium 1, however, the interface is the next-to-last line in that medium while thefictitious line is the last line. In medium 2, the interface is the first line of the z-gridin that medium. See Figure 5a. We now illustrate the procedure for equation (9).Replace the r-derivatives with a centered (two-sided) finite difference approximationand the z-derivatives with an asymmetrical (one-sided) finite difference
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