Substituting {} in Equation 65 by Equation 67, Equation 65 becomes

[ 1 ] η} [ 2 ] ηB } [ 4 ]K 3 ] [ 4 ] {ηΓ}= {Q 5} [ 4 ]K 3 ] {Q 6 }

K {$ - K {$ - K [

- K [

-1

-1

T

$

K

(68)

Finally, after proper assembling, we have

[ ] η} {f }

A {$ =

(69)

Equation 69 is the desired linear system of equations, which is the finite-element

representation of the mild-slope equation for open sea problems. Notice that the boundary

conditions, including coastline boundaries and the circular open boundary, are all

consolidated in Equation 69. The solution method used in CGWAVE for Equation 69 is

described in the next section.

The finite-element formulation given above is for open-sea offshore problems. In

case of harbor problems, the formulation is analogous. The only difference arises from the

treatment of the open boundary condition. The classical treatment of these problems

assumes that the coastlines outside the model domain are straight, collinear and fully

reflective. The exterior wave field is written as η ext = η I + η R + η s, where η I , η R, and

$

$

$

$

$

$

η s represent the incident, the reflected, and the scattered wave fields, respectively. Based

$

on the assumptions, we define (Demirbilek and Gaston 1985)

η0 = ηI + ηR

$

$

$

= Ae ikrcos(θ-θI ) + Aeikrcos(θ+ θI )

(70)

∞

= 2A ∑ εn i n J n (kr)cosnθI cosnθ

n=0

where A is the incident wave amplitude and θI is the incident wave angle with respect to

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