Page 123 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

114 Fundamentals of Ocean Renewable Energy

5.1.4 Irregular Waves
So far, we have considered a wave that is characterized by a single frequency and
direction (i.e. a monochromatic wave). Monochromatic waves can be generated
in laboratories (wave tanks), but are rarely seen in the ocean. Waves, which have
been generated by wind, are irregular and include many wave frequencies. As
‘wind waves’ travel over long distances, they are dispersed, because the wave
celerity/speed is dependent on the frequency. These waves are called ‘swell
waves’ and are more similar to regular waves.
Within the range of validity of linear wave theory (see the next section),
we can superimpose several monochromatic/sine waves to construct a more
realistic irregular wave. Conversely, we can decompose an irregular wave into a
number of sinusoidal waves, which have different frequencies. Mathematically
speaking, this is the concept of the Fourier series/integral, which is based on the
principal that any function can be constructed by combining a set of simple sine
waves. This is a powerful method, because we can also find the wave properties,
such as pressure, acceleration, and energy, by combining/superimposing the
hydrodynamic field of these individual sine waves.
Two main approaches exist to deal with irregular waves: statistical and
spectral. In the statistical approach, statistical variables are calculated and used
to represent irregular waves. For instance, parameters such as the average wave
period, and the root mean square wave height, can be used to represent the
wave period, and the wave height, respectively, for a realistic sea state. In this
approach, the sea state is represented by a probability distribution function such
as Rayleigh, and statistical parameters are derived from that distribution. The
spectral method is based on the concept of the Fourier transform and is the basis
of spectral wave models, such as SWAN [2], TOMAWAC [4], or STWAVE [5],
which are commonly used for wave energy resource characterization. A detailed
discussion about irregular waves is beyond the scope of this book and can be
found in many other resources (e.g. [1,6]). Here, basic concepts, with a focus on
wave power, are presented.
As mentioned, we can combine several waves with a range of frequencies
and phases, to construct an irregular wave, or vice versa. Mathematically
speaking, this means
N

η(t) = a i sin(σ i t + g i ) (5.22)
i=1
where a i and g i are the amplitude and phase of an individual sine wave,
respectively, and η represents the time series of an irregular wave. The total
energy of this irregular wave is found by adding up the energy of all sine waves
as follows
N N
1 2 1 2
E = ρg a = ρg H i (5.23)
i
2 8
i=1 i=1