α2 1

3

(41)

2 6

where

∑a

cos *k * j ⋅ ( *x*g - *x*r ) - ω jt + ε j

η( *x*g , *t *) =

(42)

(43)

να*xy *( *x*g , *t *) = -∑ a j k 2 sin 2 θ cos θ cos *k * j ⋅ ( *x*g - *x*r ) - ω j t + ε j

(44)

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 (*HL*2/*h*3) 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.

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, *S*η(ω,θ), which is the product of the fre-

quency spectrum, *S*η(ω), and a directional spreading function *D*(ω,θ):

=

(45)

The directional spreading function is non-negative and should satisfy the follow-

ing relation:

π

∫

=

1

(46)

-π

1

A useful guideline is *H*mo/*L*p > 0.025 tanh *k*ph, where *H*mo is the significant wave height, and *L*p

spectrum.

19

Chapter 3 Numerical Solution

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