α2 1
u f ( xg , t ) = [h + η( xg , t )]uα ( xg , t ) + h 2 - (uαxx ( xg , t ) + vαxy ( xg , t ))
3
(41)
2 6
where
N
∑a
cos k j ⋅ ( xg - xr ) - ω jt + ε j
η( xg , t ) =
(42)
j
j=1
N
uαxx ( xg , t ) = -∑ a j k 2 cos3 θ cos k j ⋅ ( xg - xr ) - ω jt + ε j
(43)
j
j =1
N
ναxy ( xg , t ) = -∑ a j k 2 sin 2 θ cos θ cos k j ⋅ ( xg - xr ) - ω j t + ε j
(44)
j
j=1
conditions would have to be modified to take into account the presence of lower
and higher frequency wave components induced by nonlinear interactions
between the primary wave components. Second-order Boussinesq theory can be
used to generate the velocities and flux densities associated with the bound
second-order waves along the wave generation boundary. However, this theory is
valid over a limited range of wave steepnesses (H/L) and relative depths (h/L)
because it only includes second-harmonic terms. Second-order Stokes theory, for
example, cannot accurately describe the shape of cnoidal-type waves in shallow
water when the Ursell parameter (HL2/h3) is large.
A different approach to generating steep irregular waves in shallow water is
the nonlinear Fourier method of Osborne (1997), in which an irregular wave train
is represented as a superposition of nonlinear cnoidal or solitary type waves. This
approach, however, has not yet been implemented in BOUSS-2D.
Irregular Multidirectional Waves. Naturally occurring ocean waves exhibit
a pattern that varies randomly not only in time but also in space. The wave
energy is distributed over both frequency and direction and can be described in
terms of a directional wave spectrum, Sη(ω,θ), which is the product of the fre-
quency spectrum, Sη(ω), and a directional spreading function D(ω,θ):
Sη (ω, θ)
=
Sη (ω)D(ω, θ)
(45)
The directional spreading function is non-negative and should satisfy the follow-
ing relation:
π
∫
D(ω, θ)
=
1
(46)
-π
1
A useful guideline is Hmo/Lp > 0.025 tanh kph, where Hmo is the significant wave height, and Lp
spectrum.
19
Chapter 3 Numerical Solution
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