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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