where
{(
) }
)
(
i ~
{Q 5} =
k a A q N Γ + q 1 L1 , (q 1 + q 2 )L2 , ... , q N Γ- 1 + q NΓ LNΓ
T
(58)
2
and
[
]
q P = cos(θP - θI )exp ikR cos(θP - θI ) ,
P = 1, 2, ... , NΓ
(59)
The last integral I6 in Equation 45 involves both η S and η I. Here, the Bessel-
$
$
Fourier form of η I (see Equation 15) will be used and the integral can be found
$
analytically:
η
$
I6 = ⌠ ~ηI S ds
a$
⌡G
n
m
m
⌠
= k~A ∑ εn i n J n (kr)cosn(θ - θI )∑ H'n (α n cosnθ + βn sinnθ)ds
a
⌡ Γ n =0
(60)
n =0
2π m
⌠
m
J n (kr)cosn(θ - θI )∑ H 'n (α n cosnθ + βn sinnθ)Rdθ
∑εi
= k~A
n
a
n
⌡0
n =0
n =0
= {Q 6 }{}
T
where
{Q 6 } = 2πkR~A
T
a
(61)
{J H , iJ H cosθ , iJ H sinθ , ... , i
}
J m H cosmθI , i J m H sinm θI
'
'
'
m
'
m
'
0
0
1
1
I
1
1
I
m
m
Now, we have evaluated all the integrals of the functional J defined by Equation 19
using a linear triangular element network. Collecting these integrals together, we have
24
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