2 BASIC EQUATIONS
The solution of the two-dimensional elliptic mild-slope wave equation is a well-
accepted method for modeling surface gravity waves in coastal areas (e.g. Chen &
Houston, 1987; Chen, 1990; Xu & Panchang, 1993; Mei, 1983; Berkhoff, 1976; Kostense
et al., 1986; Tsay and Liu, 1983). This equation may be written as:
Cg
(
)
∇ ⋅ CCg ∇ η +
σ2 η = 0
$
$
(1)
C
where
η(x, y) =
complex surface elevation function, from which the wave
$
height can be estimated
σ
=
phase velocity = σ/k
C(x,y)
=
group velocity = σ / k =nC with
Cg(x,y) =
1
2kd
n=
1 +
(2)
2 sinh 2 kd
wave number (= 2π/L), related to the local depth d(x,y)
k(x,y)
=
through the linear dispersion relation:
σ2 = gk tanh (kd)
(3)
Equation 1 simulates wave refraction, diffraction, and reflection (i.e. the general
wave scattering problem) in coastal domains of arbitrary shape. However, various other
mechanisms also influence the behavior of waves in a coastal area.
The mild-slope
equation can be modified as follows to include the effects of frictional dissipation
(Dalrymple et al 1984; Chen 1986; Liu and Tsay 1985) and wave breaking (Dally et al
1985; De Girolamo et al 1988):
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