Page 127 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

118 Fundamentals of Ocean Renewable Energy

Finally, we can find out how the energy wave period can be calculated based
on a wave spectrum. By comparing Eqs (5.36), (5.38),

gT E 1 2 m −1
P = (ρgm 0 ) = ρg m −1 → T E = 2π (5.40)
4π 2 m 0
which is equivalent to T m−10 in spectral models such as SWAN.
In brief, wave power can either be computed using the wave spectrum (i.e.
Eq. 5.32) or by using a simplified equation based on the statistical wave pa-
rameters (energy/average wave period and significant wave height; Eq. (5.39)).
The simplified method is based on the deep water wave approximation, which
is generally valid in the vicinity of WECs. The range of water depths where a
WEC is installed depends on the device (40 m is a typical depth). Consider a
wave with a period of 8 s, propagating in 40 m water depth. The wave length is
(from the dispersion equation) L = 2π/k = 99 m, and therefore kd = 2.54 <π;
for this wave, tanh(kd) = 0.98 which is close to 1 and hence, the deep water
approximation is valid.

5.1.5 Nonlinear Waves

Linear wave theory is the basis of many analytical methods and numerical
models, which describe the properties and propagation of regular/irregular
waves in the open ocean and coastal regions. Nevertheless, we should be aware
of the range of validity of this theory, and wave processes which cannot be
explained by this theory. Referring to Section 5.1.1, several assumptions were
made to develop linear wave theory, which may not be valid in real-word
applications. In particular, in the derivation of linear wave theory, it is assumed
that the amplitude of the wave is small compared with the water depth and
wave length. Fig. 5.4 shows the range of validity of linear wave theory. As

you can see, for deep waters e.g. d > 10 −3 and small amplitude waves
gT 2
H −3
e.g. < 10 , linear wave theory is valid; however, in shallow waters and
gT 2
for large amplitude waves, other wave theories such as Stokes and Cnoidal are
more appropriate. Note that in certain depths, or for certain wave steepnesses,
the wave is no longer stable and will break. This is discussed further in the
following section, and is shown by breaking criteria on this figure (shallow and
deep criteria).
Wave Breaking
When the wave height, and consequently wave steepness (i.e. the ratio of
wave height to wave length) increase, the water surface profile deviates from
a sinusoidal wave shape. The wave eventually breaks if the wave steepness
becomes too high.