Geoscience Reference
In-Depth Information
The following chain of equalities gives ˛.a;R/.
J
D
˛
fm
.n
1
n
C
/
D
˛
fm
.a;R/.n
R
n
C
/
D
˛
fm
.a;R/Œn
1
.n
1
n
C
/b
.R/
n
C
D
˛
fm
.a;R/.n
1
n
C
/Œ1
b
.R/
or
˛
fm
b
C
.R/
˛
fm
.a;R/
D
(3.48)
From Eqs.
3.28
and
3.44
we find:
D
S
p
a
2
v
T
;
˛
fm
(3.49)
Now we must express n.r/ via n
R
instead of
n
1
.Wehave
n
fm
.r/
D
n
1
b
C
.r/
C
n
C
b
.r/:
(3.50)
From Eq.
3.50
we have
n
fm
.R/
D
n
1
b
C
.R/
C
n
a
b
.R/:
(3.51)
We solve this equation with respect to
n
1
and find
n
R
n
a
b
.R/
b
C
.R/
n
1
D
(3.52)
Now
n
R
n
C
b
.R/
b
C
.R/
n.r/
D
b
C
.r/
C
n
C
b
.r/:
(3.53)
Next,
n
R
n
C
b
.R/
b
C
.R/
n
R
n
C
C
n
C
n
a
b
.R/
b
C
.R/
n
R
n
C
b
C
.R/
C
n
C
:
D
D
(3.54)
And finally,
n
R
n
C
b
C
.R/
b
C
.r/
C
n
C
:
n.r/
D
(3.55)
In deriving this equation we used the identity b
C
.r/
C
b
.r/
D
1.
