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Aanderaa Data Instruments AS – TD302
Wave Spectrum
:
The magnitude of one frequency component of the power spectrum is proportional to the square of the amplitude
of the signal with the very same frequency. That is, if the wave is a pure sine wave with amplitude
a
, the magnitude
of the spectrum component with the same frequency as the sine wave, is
a
2
. The wave energy is given as:
2
2
8
2
gH
ga
E
ρ
ρ
=
=
where
ρ
is the medium density,
g
is the gravitational coefficient and
H
is the wave height. The wave spectrum
calculated by the WTS are scaled in this manner.
Spectral Moments
:
A number of
spectral moments
are calculated based on the wave spectrum. Next, these spectral moments are used
to compute wave parameters. The spectral moments are defined as follow:
0
( )
N
n
n
i
i
i
m
f S f
=
=
∑
where
f
i
is a specific frequency and
S(f
i
)
is the wave spectrum component at that frequency. Hence, the moment of
order zero,
m
o
, is found by summing the wave spectrum components from zero frequency up to the cut-off
frequency.
Based on the above considerations and the definition of the spectral moments, the moment of order zero is equal to:
2
0
0
2
N
i
i
a
m
=
=
∑
Assuming that
X
is a simple sinusoidal wave with amplitude
a
, the sampled values
X
i
are:
sin( )
i
i
X
a
y
=
The variance,
σ
, of
X
is:
2
2
2
2
(
)
(
sin( ))
2
i
i
a
E X
E a
y
σ
µ
=
−
=
⋅
=
where
E
denotes the expectation and
μ
is the average of X (
μ
=0 for an ideal sine wave). The wave motion can be
modelled as a superposition of simple sinusoidal waves. Hence, the individual spectral components in the wave
spectrum can be seen as the variance of individual sinusoidal waves with different frequencies. The zero order
spectral moment,
m
0
, which then is the sum of the variances of the individual spectral components, can be seen as
the total variance of the wave record.