For simplicity, the open boundary Γ is assumed to be a circle of radius R. For

computational purposes, the series for the scattered waves (Equation 12) and the series

for incident waves (Equation 15) are truncated after a finite number of terms. In principle,

trial and error should be performed in modeling a certain case in order to choose the

appropriate number of terms. Here, we assume that the series will be truncated after m

terms.

By using the orthogonality of trigonometric functions:

when n ≠ m

0

2π

∫

sin nθsin mθdθ =

(46)

π

when n = m

0

0 when n ≠ m

2π

∫

cos nθcos mθdθ = π when n = m ≠ 0

(47)

0

2π when n = m = 0

and substituting η S with Equation 12, the line integral I3 in Equation 45 can be evaluated

$

analytically, as follows

π

m

∑(

)

I 3 = kR~ 2α 2 H 0 H '0 +

α 2 + β2 H n H 'n

(48)

a

0

n

n

2

n=1

where k and ~ can be taken as average values along Γ and

a

d

H n ≡ H (n1) ( kR),

H 'n ≡

H (n1) ( kr)

(49)

d( kr)

r = R

For convenience of mathematical manipulation later on, we define the following vector for

the unknown coefficients αi and βi :

21

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