Geology Reference
In-Depth Information
Recapitulation. Key implications may have well been buried behind the
facade of subscripts. Thus, it is convenient to introduce the simplified notation
P= x
(1.205)
q= y
(1.206)
r= z
(1.207)
s= t
(1.208)
so that Equation 1.181 becomes
H(
,p,q,r,s,x,y,z,t) = 0
(1.209)
The implied recipe for solving the partial differential equation in Equation
1.181 calls for the equivalent solution of a system of coupled first-order
ordinary differential equations in the nine “independent” unknowns shown in
the argument of Equation 1.209, namely,
dx/d
=H p
(1.210)
dy/d
=H q
(1.211)
dz/d
=H r
(1.212)
dt/d
=H s
(1.213)
dp/d
=- H x - p H
(1.214)
dq/d
=- H y - q H
(1.215)
dr/d
=- H z - r H
(1.216)
ds/d
=- H t - s H
(1.217)
d /d
=p H p + q H q + r H r + s H s
(1.218)
where Equation 1.218 is obtained by simplifying Equation 1.183 with the
expressions in Equations 1.210 to 1.213. Again, the p, q, r, s, x, y, z, t and
derivatives of H( ,p,q,r,s,x,y,z,t) are regarded as independent partial derivatives.
Since H is known, the expressions for the right sides of Equations 1.210 to 1.218
are explicitly available.
When initial values for , p, q, r, s, x, y, z, and t are prescribed, the solution
can be obtained by the simultaneous integration of nine first-order ordinary
differential equations in the time-like coordinate (simpler H functions, of
course, will reduce the number of coupled equations). In our subsequent study
of geophysical applications, this integration is carried out for the “eikonal
equation” ( / x) 2 + ( / y) 2 + ( / z) 2 = 1/c(x,y,z) 2 corresponding to the three-
dimensional model 2 P/ t 2 - c 2 ( 2 P/ x 2 + 2 P/ y 2 + 2 P/ z 2 ) = 0 for normal
stresses in the earth.
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