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∞
∞
d
2
dy
2
Z
S
1
R
2
(
y
)
a
n
cos
nly
−
a
n
cos
nly
+
i
o
h
(
y
)
=
,
(8
.
8)
1
1
where
S
1
=
h
1
/
1
and
R
2
(
y
)
=
h
2
(
y
)
2
.
On differentiation, we write
1
2
h
o
a
n
n
2
cos
nly
∞
∞
l
2
S
1
2
a
n
n
2
{
cos (1
+
n
)
ly
+
cos (1
−
n
)
ly
} −
h
2
1
1
∞
Z
−
a
n
cos
nly
−
i
o
h
o
cos
ly
=
−
i
o
(
h
1
+
h
2
)
.
1
9)
Fourier coefficients
a
n
are found by minimizing the misfit of (8.9). The arith-
metic shows that we can restrict ourselves to series with 7 terms. In the case that
h
o
≤
(8
.
0
.
3
h
2
and
L
≥
4
h
o
, the low-frequency transverse impedance is approximated
by formula
Z
⊥
(
y
)
≈
o
h
⊥
(
y
)
where
h
2
−
⊥
h
o
cos
2
L
h
⊥
(
y
)
=
h
1
+
y
.
(8
.
10)
⊥
is a distortion factor dependent on the
galvanic ratio
⊥
=
Here
L
/
d
min
,
where
d
min
is the minimum adjustment distance:
e
−
3
/
(
⊥
)
1
.
4
⊥
=
S
1
R
min
(8
.
11)
R
min
2
h
min
2
d
min
=
,
=
2
.
2
A similar approximation can be proposed for the TE-mode. Let us consider the
low-frequency longitudinal impedance. Reducing (7.46) to the
h
-interval, we get
h
1
h
2
(
y
)
d
2
E
x
(
y
)
dy
2
−
E
x
(
y
)
=
i
o
h
(
y
)
H
y
(
y
)
.
(8
.
12)
Obviously
E
x
(
y
) and
H
y
(
y
) are even periodic functions with the period
L
.They
can be represented by the Fourier decompositions
b
n
e
nlz
cos
nly
z
=
0
=
∞
1
∞
1
E
x
+
E
x
+
E
x
(
y
)
=
b
n
cos
nly
z
=
0
=
(8
.
13)
∞
1
o
1
E
x
(
y
,
z
)
l
H
y
+
H
y
(
y
)
=
b
n
n
cos
nly
,
i
z
i
o
where
E
x
=
Z
H
y
and
H
y
are normal fields obtained for
h
o
=
0.
