spectra.

Expressions for the different wave spectra are provided in Appendix B.

The Fourier series technique is used to generate time-histories of the velocity

boundary conditions from the wave spectrum. The water-surface elevation, η(*t*),

is assumed to be a zero-mean, stationary, random Gaussian process. The surface-

elevation time series at a reference point *x*r = (*x*r, *y*r) in the computational domain

can be represented as a linear superposition of *N *regular wave components, i.e.,

∑a

cos *k * j ⋅ *x*r - ω j t + ε j

η( *x*r , *t *) =

(36)

phase angle, and wave number vector of the *j*th frequency component, respec-

tively. The angle, θ, is the direction of wave propagation relative to the positive

x-axis.

The wave spectrum is divided into *N *frequency bands with uniform spacing,

amplitudes of the individual wave components are obtained deterministically

from the wave spectrum, *S*η(ω), as:

2*S*η (ω j )∆ω

(37)

while the phase angles, εj, are randomly selected from a uniform distribution

between 0 and 2π. Incident wave conditions are more typically specified in terms

of the repeat period or duration of the record, *T*D, and time-step, ∆*t*. The values of

1

∆ω =

(38)

(39)

2∆*t*

The velocity and flux boundary conditions along a wave generation line perpen-

dicular to the x-axis may be obtained from the surface elevation using the linear

transfer function approach:

∑T (ω )*a*

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

(40)

18

Chapter 3 Numerical Solution

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