PROPAGATION OF ELASTIC WAVES IN LAYERED MEDIA BY ...

370 BISLLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICAeq. (1) may be writtenar~ + r Or r" + arazj + \ L Oz. arazj v3 at ~(~)az~ +r-~ + azarj + ~ L ar2O2A 10A 1 OB] _ 1 O2B (4)azar r az + r-~:r J vo 2 at 2whereVc = [(X -~- 2tt)/p] 1/2vo = [E41/2. (5)v~ is the eompressional speed and v.~ is the shear speed of the medium. In the discussionthat follows, when superscripts 1 and 2 are attached to the quautities A andB, they refer to media 1 and 2, respectively. Elastic velocities in the two mediaare distinguished by subscripts.The boundary conditions for the problem are determined by the fact that thestresses must vanish on the free surface, and that both stresses and displacementsmust be continuous at the interface. The normal and tangential stresses across theplane surface z - const, can be expressed in terms of the displacement components asOA 1 A + OBT== (X+2~) -~:r -t-r ~zz --2~ n+-~lab OA] (6)~.=~ ~:r+-~z •Hence the boundary conditions at the free surface z = 0 are given byoA 1 A1 0B:l (Vsl~2 [I A I oA1 l~:-r +- + az j - 2 -- + =0r \v~/ Or JThe boundary conditions at the interface z = h are given by[OA 1 1A~ abel_ (v~2[!A~ OA11Or + - + az j 2 +r \v~:! Or Jar + az J O. (7)\yell L ar + 7 + az j 2 Pll \Vcl/ ar Jar + az J ~ L ar + az J (9)