the exterior coastlines as shown in Figure 1. The scattered wave potential η s in the

$

exterior region must take the following form in order to comply with the exterior coastline

boundary conditions:

∞

ηS = ∑ H n (kr)α n cosnθ

$

(71)

n =0

as shown in Xu, Panchang and Demirbilek (1995). The corresponding functional for

harbor problems has the same form as Equation 19 except that η I in Equation 19 has to

$

be replaced by η 0 (Equation 70), η S takes the new form given by Equation 71, and the

$

$

open boundary Γ represents the semicircle as shown in Figure 1. The finite-element

formulation of harbor problems can now readily be found in a manner similar to the open-

sea problems described above, by replacing η I and η S with Equation 70 and Equation 71

$

$

For most practical cases, the fully reflective straight coastline assumption in the

classical treatment of the open boundary condition is improper and the effects may

substantial. Xu, Panchang and Demirbilek (1995) have shown that it is preferable to use

the parabolic approximation (Equation 13) as the open boundary condition. For harbor

problems, along the open boundary Γ (Figure 1) we use (13) as the boundary condition

for the scattered waves. Matching the potential and its normal derivative along Γ and

using the parabolic open boundary condition (13), we have the total potential as

η = η0 + ηS

(72)

$

$

$

and

η

ηS

η

η

ηS

2

$

$

$

$

$

= 0+

= 0 - pηS + q

(73)

$

θ2

r

n

r

r

27

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