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 :
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