Actions on wind turbines - Concrete Structures for Wind Turbines - page 35

2.5.3 Harmonic primary wave
The harmonic primary wave (Airy wave) is a solution to the linear, or linearised,
boundary value problem for the propagating gravity wave, that is for a wave whose
restoring force is the force of gravity. This solution is linked with the fundamental
assumption of an infinitesimal amplitude, or rather steepness ( ¼ wave height/wave-
length). The profile of a long-crested (smooth) harmonic primary wave that propagates
in the direction of the x axis over a constant depth of water (d) can be described by the
following formulation (Figure 2.20):

H
2 cos k x v t

z ðÞ¼ x ; ð Þ

where

H wave height
k ¼ 2 p / l wavenumber
l wavelength
v¼ 2 p /T angular frequency
T wave period

The flow field of the primary wave can be unambiguously described by the velocity
potential F (x, z, t), which is

H
2

g
v

cosh k ½ ð z þ d Þ

F ð x ; z ; t Þ¼ sin k x v t ð Þ

cosh k d ð Þ

for the wave profile z (x, t).

The most important feature of the Airy wave is the existence of a dispersion, that is the
angular frequency's dependence on the wavelength or wavenumber, coupled with the
gravitational acceleration (g) as an indication of the gravity wave. The dispersion
equation is usually written in the following form:

2

v ¼ g k tanh k d ð Þ

Fig. 2.20 Harmonic primary wave (H¼10.0m, d¼30m, l¼150m)