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

Integrated Publishing, Inc. |