Wind Energy

10 L. Bergdahl et al.
2.4 Wave Kinematics
The wave kinematics can be realized to second order for weakly non-linear
waves in deep and shallow water. Equation (2.2) and (2.3) give the first and
second order contributions. F (n) are the first-order component amplitudes, ωn
the angular frequencies and kn the wave numbers. H(ωn, ωm) is a quadratic
transfer function. For details see [7].
1η(x, t) =
N�
F (m) exp i(ωmt − kmx) (2.2)
m=0
N� N� F (n)F (m)
2η(x, t) =
H(ωn,ωm)expi[(ωn + ωm)t − (kn + km)x]
4g
n=0 m=0
(2.3)
The equations are valid to the mean water elevation and have to be extrapolated
to the instantaneous water surface.
2.5 Example of Wave Loads
In Figs. 2.3 and 2.4 comparisons of linear (1st order) and non-linear (1st+2nd
order) realizations of forces and moments are shown [4]. For these high waves
Force (10 3 N)
150
100
50
0
−50
−100
1900
Force, comparison between linear and non-linear realisation
1910 1920 1930 1940 1950 1960
Time (s)
Linear
Non-linear
1970 1980 1990 2000
Fig. 2.3. Comparison of linear and non-linear force