As shown in Figure 2, the entire domain is
Γ∞
separated into two sub-domains. Domain Ω is the
Ω0
numerical model domain. Domain Ω 0 is the exterior
Γ
Ω
domain extending to infinity. We assume that
B
complicated topography, structures, and islands, are
located inside the circular boundary Γ (in domain Ω ).
In Ω 0, the total wave potential can be written as the
sum of incident wave potential and the scattered
Fig. 2 Definition sketch
wave potential:
ηext = ηI + ηS
(17)
$
$
$
For brevity, we write the governing Equation 4 in the general form:
~
∇ ⋅(~∇ η)+ b η = 0
(18)
a $
$
~ ≡ CC and ~ ≡ C g σ2 + iσw + iC σγ.
b
where a
g
g
C
Mei (1983) has shown that the problem of solving Equation 18 with boundary
conditions described by (9) on coastlines/structures and by Equation 11 at infinity is
equivalent to the stationary of the following functional J:
[
]
1
1
~
J = ⌠ ⌠ ~(∇ η) - b Φ 2 dA - ⌠ α ~η2ds +
2
a $
a$
⌡⌡ 2
⌡B 2
Ω
(19)
⌠ ~ 1
( ηS + ηI )
η
$ S + ηI S - η
$
$
$
a 2 η
$
ds
$
n
n
⌡Γ
The solution of the wave potential can be found by minimizing J over domain Ω .
13
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