P

Γ

where { η } is the subset of { η } for

$

$

nodes situated on boundary Γ :

θP

Ω

{ } = {η , η , η

}

T

ηΓ

Γ

Γ

Γ

, ... , ηΓΓ

$

$ $ $

$N

1

2

3

1xNΓ

(55)

Γ

and

[K4]

is

a

fully

populated

N Γ M matrix :

Figure 5. Definition sketch for line integrals

along the open boundary Γ

k~

a

[

K 4 ]=

2

(

)

(

)

H 'n sin nθN Γ- 1 + sin nθ1 L1 ⋅ ⋅

2H '0 L1 ⋅ ⋅ H 'n cos nθN - 1 + cos nθ1 L1

⋅

⋅

Γ

2H 0 L ⋅⋅ H n (cos nθ1 + cos nθ2 )L

H n (sin nθ1 + sin nθ2 )L

⋅⋅

⋅

⋅

' 2

'

2

'

2

⋅

⋅

⋅

⋅

⋅

⋅

⋅

⋅

⋅

⋅

⋅

⋅

⋅

⋅

⋅

' NΓ

(

)

(

)

2H 0 L ⋅⋅ H 'n cos nθN Γ - 1 + cos nθN Γ LN Γ H 'n sin nθNΓ - 1 + sin nθNΓ L2 ⋅ ⋅

⋅

⋅

(56)

where n = 1, 2, ..., m.

The next integral in Equation 45 is I5. Similar to the treatment in I4, we will take

the center value of η I/ n and assume linear variation of η for a segment on boundary Γ.

$

$

Substituting η I in I5 by Equation 15, it is easy to find

$

η

$

I5 = ⌠ ~η I ds

a$

⌡Γ

n

i ~ NΓ P P

(

)

[

]

= k a A ∑ L ηi + ηP cos(θP - θI )exp ikRcos(θP - θI )

(57)

$

$j

2

P=1

{}

= {Q 5} ηΓ

T

$

23

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