Geoscience Reference
In-Depth Information
Taking the Bessel transform of Eq. (
5.139
) we get
@
z
A
2
A
D
0;
(5.140)
where A
D
A.k;
z
;!/ and as before
2
D
k
2
i
0
!
g
stands for the propagation
factor in the ground. We mention in passing that Eq. (
5.140
) is analogous to
Eq. (
5.113
) for the scalar potential ‰ in the ground. Taking into account that A
should tend to zero when
z
!1
we chose the solution of Eq. (
5.140
) in the form
A
D
A.
d/ exp Π.
z
C
d/;
(5.141)
where Re>0and A.
d/ denotes the value of potential A on the ground surface.
The components ıB
'
and E
r
must be continuous at
z
D
d whence it follows
that A and dž must be continuous at
z
D
d. Taking into account Eqs. (
5.132
)
and (
5.138
)forA and dž, the boundary condition on the ground surface takes the
following form
i!"
0
g
@
z
A.
d
C
0/
D
@
z
A.
d
0/:
(5.142)
Taking into account the boundary conditions at
z
D
d and combining Eqs. (
5.137
)
and (
5.141
) gives a set of algebraic equations for undefined constants. These
equations can be solved for C
1
and C
2
to yield
1
exp .kd/;
A.
d/
2
i!"
0
k
g
C
1
D
(5.143)
1
C
exp .
kd/:
A.
d/
2
i!"
0
k
g
C
2
D
(5.144)
Substituting Eqs. (
5.143
) and (
5.144
) into Eq. (
5.137
) yields the solution of the
problem in the neutral atmosphere
A
D
A.
d/
f
cosh Œk.d
C
z
/
sinh Œk.d
C
z
/
g
0
m.!/.
z
0
/
2k
sinh
k
z
0
:
(5.145)
Here we introduce the dimensionless parameter
D
i!"
0
=
g
k
. Substituting
Eq. (
5.145
)forA into Eq. (
5.132
), we obtain
A.
d/
f
sinh Œk.d
C
z
/
cosh Œk.d
C
z
/
g
kc
2
i!
dž
D
cosh
k
z
0
:
0
m.!/.
z
0
/
2k
(5.146)
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