Using Equation 72 to eliminate η s yields

$

η

η

2

$

$

= - pη

$- q 2+ g

(74)

θ

n

where

η

η0

2

$

$

g = 0 + pη0 + q

$

(75)

θ2

r

Equation 74 is the open boundary condition in terms of total wave potential η .

$

For this parabolic boundary problem, the desired Jacobian functional may be more

complicated than that of Equation 19. Therefore the Galerkin finite-element formulation is

used.

According to the Galerkin approach, the wave potential η is approximated by

$

n

η = ∑ ηi N i

(76)

$

$

i=1

$

$

node i. The unknown η i can be determined from the orthogonality conditions between

$

function Ni and left-hand side of Equation 18, that is

(

)

~

∇ (~∇ η)+ b η N i dΩ = 0

(i

= 1, ..., N )

∫

∫

(77)

a $

$

Ω

Using the divergence theorem, Equation 77 becomes

28

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