Geoscience Reference
In-Depth Information
As a starting point, we use Eq.
3.115
rewritten in terms of the dimensionless
variables:
Z
e
u
2
".a/
D
.
u
/d
u
:
(3.151)
0
It is possible to prove that .
.
u
// has no zeros at
u
> 0and .
.0//
D
0.
Equation
3.149
confirms this statement.
At small
a
(large ), small
u
/
1= contributes to the integral on the right-
hand side of Eq.
3.151
. We thus must seek the roots of Eq.
3.143
in the limit of
small
u
. As is seen from Eq.
3.148
,small
u
corresponds to large . The following
approximations are used:
1
"
C
1
S./
1
"
C
2
1
4
C
2
2"
C
3
1
6
and
1
1
4
C
2
6
:
1/
2
.
2
Now we get
"
C
1
"
C
2
1
4
C
4
"
C
1
2"
C
3
1
6
u
D
(3.152)
In the lowest approximation we have
s
"
C
1
."
C
2/
u
2
.
u
/
To find the next approximation, we rearrange Eq.
3.152
:
s
"
C
1
."
C
2/
u
s
1
C
4
"
C
2
2"
C
3
1
2
2
D
The last term under the square root on the right-hand side of this equation is
small. Expanding the square root and replacing
2
with its lowest approximation
give
s
"
C
1
."
C
2/
u
"
C
2
2"
C
3
C
2
.
u
/
D
2
