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where C
3
and dž.0/ are undetermined coefficients. In order to match the
solutions (
5.15
) and (
5.101
) at the boundary
z
D
L, one should take into account a
requirement of continuity of the potential dž and its derivative @
z
dž. Whence we get
C
3
sin x
0
C
dž.0/ cos x
0
D
C
1
exp .ix
0
/;
(5.102)
C
3
cos x
0
dž.0/ sin x
0
D
iC
1
exp .ix
0
/:
(5.103)
where x
0
D
!L=V
AI
denotes the dimensionless frequency and
D
V
AI
=V
AM
.The
set of Eqs. (
5.102
)-(
5.103
) can be solved for C
3
to yield
C
3
D
idž.0/
1
C
.1
/ exp .2ix
0
/
1
C
C
.1
/ exp .2ix
0
/
:
(5.104)
Substituting Eq. (
5.104
)forC
3
into Eq. (
5.101
), we come to Eq. (
5.18
), which
describes the potential dž inside the IAR region.
Similarly, the solution of Eq. (
5.13
) describing FMS waves in the region 0<
z
<
L can be written as
‰
D
‰.0/ cosh
I
z
L
C
C
4
sinh
I
z
L
;
(5.105)
where the function
I
is given by Eq. (
5.20
). As before C
4
and ‰.0/ denote
undetermined coefficients. On account of the continuity of the potential ‰ and its
derivative @
z
‰ at the boundary
z
D
L we get
‰.0/ cosh
I
C
C
4
sinh
I
D
C
2
exp
M
;
(5.106)
‰.0/
I
sinh
I
C
C
4
I
cosh
I
D
C
2
M
exp
M
;
(5.107)
where the function
M
is given by Eq. (
5.17
). The set of Eqs. (
5.106
)-(
5.107
) can
be solved for C
4
to yield
C
4
D
‰.0/
I
C
M
.
I
M
/ exp .2
I
/
I
C
M
C
.
I
M
/ exp .2
I
/
:
(5.108)
Substituting Eq. (
5.108
)forC
4
into Eq. (
5.105
), we come to Eq. (
5.19
), which
describes the potential ‰ inside the IAR.
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