where Hn(kr) are the Hankel functions of the first kind. The Hankel functions of the
second kind do not satisfy the Sommerfeld radiation condition at infinity and are hence
excluded from (12).
However, the ηS given in (12) requires that the exterior domain be of constant
depth. Also for harbor problems (Figure 1), the scattered wave potential as described by
(12) demands straight, collinear and fully reflective coastlines in the exterior region. To
overcome these problems, Xu, Panchang and Demirbilek (1996) have developed an
alternative scheme in dealing with the open boundary condition. This consists of using
the following parabolic approximation along the open boundary :
ηS
ηS
$2
$
+ pηS + q 2 = 0
$
(13)
θ
r
where
1
k 2 r 2 + k 2 r 2 + ik 0 r +
1
0
4
and q =
p=
(14)
2ik 0 r 2
2ik 0 r 2
In Equation 14, k0 can be taken as the wave number corresponding to the averaged water
depth along the open boundary Γ. Within the model domain Ω , the mild-slope equation
applies. The parabolic approximation (13) will be used only along the semi-circular arc Γ
as the open boundary condition. The actual implementation of these boundary conditions
is described later.
10
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