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