REFLECTION AND TRANSMISSION OF SEISMIC WAVES UNDER INITIAL STRESS AT THE EARTH ’ S CORE-MANTLE BOUNDARY

In the present paper the influence of the initial stress is shown on the reflection and transmission of P waves at the core-mantle boundary. Taking a particular value of the inherent initial stress, the variations of reflection and transmission coefficients with respect to the angle of emergence are represented by graphs. These graphs when compared with those having no initial stress show that the effect of the initial stress is to produce a reflected P and S waves with numerically higher amplitudes but a transmitted P wave with smaller amplitude. A method is also indicated in this paper to calculate the actual value of the initial stress near the core-mantle boundary by measuring the amplitudes of incident and reflected P waves.


i. INTRODUCTION.
The reflection and transmission of seismic waves at the earth's core-mantle boundary have been discussed by Dana i I, Ibrahim 2] and many other investi- gators.From their discussions we see that the reflected and transmitted waves are dependent on elastic parameters, densities and the angle of incidence.But the mantle and core contain a considerable amount of initial stress which is compressive and supposed to be hydrostatic in nature (Jeffreys,3]).The present paper shows that this initial stress has also a significant effect on the reflected and transmitted waves at the core-mantle boundary.The paper is constructed on the assumption that the core is liquid and that there is no discontinuity of initial stress at the core-mantle boundary.For simplicity, only P wave incident from the mantle side has been considered.This P wave produces reflected P (PCP), reflected S (PCS) and transmitted P waves, each of which is influenced by the initial stress.Taking a particular value of the inherent initial stress, the numerical values of reflection and transmission coefficients for different angles of emergence have been calculated and the results are given by graphs.The corresponding graphs when the initial stress is not considered are also given for comparison.From these graphs it is found that the initial stress increases the numerical values of the coefficients for the reflected P and S waves but decreases the same for the transmitted P waves.
It is shown at last that from the expression of the ratio of the coefficients for the reflected and incident P waves we may calculate the actual value of the initial stress near the core-mantle boundary.

FORMULATION AND SOLUTION OF THE PROBLEM.
Let us assume that y 0 be the boundary of the earth's core (Fig. i).The mantle and core are supposed to be homogeneous and isotropic elastic media.Let H be the initial compressive hydrostatic stress just outside and inside the core including the boundary.
The wave equations with initial hydrostatic stress are the same as those with- out initial stress (Dey, [4]).They are given by where p is the density, % and are Lame's constants and V 2 + x 2 y2 (2. lab) We shall consider only P wave incident from the mantle side.The solutions of equations (2.1ab) are A exp [ik(ct where k is the wave number, a is connected with the angle of emergence e by the relation a tan e, and a= -i b= c/ -i and For the outer core, which is supposed to be a liquid, the transmitted P wave is given by where a c2/2 1 s , % is the Lame constant and p" is the density just inside the core.
The boundary conditions require that the vertical displacement v and the incremental boundary force Af per unit initial area are continuous across the Y surface y 0 and the incremental tangential force Af per unit initial area x vanishes at the same surface.These conditions are equivalent to v v', Af 0, Af Af at y O.
x y y (2.4) The quantities without and with primes refer to the mantle and core sides respectively.
We write the displacements u, v in terms of the functions , by the relations u v -+ (2.5) x y y x Af and Af are given in Biot 5]  x y Af where sij are incremental stresses and are expressed by (Biot, [ 5]) s12 2 exy s22 s + 2 eyy (2.7) when the initial stress is hydrostatic.
Equations (2.5), (2.6) and (2.7) change the boundary conditions to Introducing the non-dimensional parameters p /8, q /e and H/2 to the relations (2.9) we obtain where + (q sec e 2 8 (p2/q2)p2 sec e tan e ) tan e with 8 /" From (2.10) it is clear that AI/A, BI/A and A'/A depend on in addition to 8, e and elastic parameters.

o
At the grazing and noral incidences i e when e and 90 respectively, /A, B1/A and A /A are independent of .When e 0 , we get AI/A -i and B1/A A /A 0. This indicates that there is a total reflection with reversal of phase at the grazing incidence.When e 90 , we get AI/A BI/A 0 and A /A 0.58.This means that no reflection of P wave occurs but a part of it is transmitted through the core-mantle boundary at the normal incidence.When e lies between 0 and 90 , increases the numerical of A1/A and B1/A but decreases A /A.When is not taken into account, AI/A attains the maximum value 0.4 near e 25.By the consideration of the maximum value of A1/A becomes 0.6 near e 20 If is omitted, AI/A equals A /A at e 34 approximately.This value of e changes to about 45 by the presence of To calculate the actual value of the initial stress, the first equation of (2.10) may be used, which is of the form A I -= f(A, p, q, 8, (3.1) where p, q and are supposed to be known quantities.The angular distance A between the epicentre and station for a surface focus is given by = dT cos e (3 2) R dA dT where R is the radius of the core (3470 Km) and has been computed from the travel times of PCP (Bullen, [6]).
Hence a careful measurements of AI/A will lead to the computation of .
ACKNOWLEDGEMENT.The authors wish to offer their sincere thanks to Professor Markus Bth, Seismological Institute, Uppsala, SWEDEN for suggesting the problem.
figure.From these graphs we infer the following things: FIG.2.