c. The JONSWAP spectrum for fetch-limited conditions in deep water.
d. The TMA spectrum for fetch-limited conditions in shallow water.
e. The Ochi-Hubble spectrum for bimodal sea states with double-peaked
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 xr = (xr, yr) in the computational domain
can be represented as a linear superposition of N regular wave components, i.e.,
N
∑a
cos k j ⋅ xr - ω j t + ε j
η( xr , t ) =
(36)
j
j=1
phase angle, and wave number vector of the jth 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, Sη(ω), as:
aj =
2Sη (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, TD, and time-step, ∆t. The values of
N and ∆ω can be obtained from these as:
1
∆ω =
(38)
TD
TD
N=
(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:
N
∑T (ω )a
cos θ cos k j ⋅ ( xg - xr ) - ω jt + ε j
uα ( xg , t ) =
(40)
u
j
j
j=1
18
Chapter 3 Numerical Solution
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