4.3 Statistical and Spectral Properties of Waves
143
Frequency analysis mainly deals with an evaluation of the distribution of wave
energy among various frequencies and directions. There are two main methods for the development of the frequency spectrum. The traditional method
is based on the Fourier Transform of the correlation function while in the second method, the time series is simply transferred into its Fourier components.
This method is known as Fast Fourier Transform (FFT). We examine these
techniques in more detail in Chap. 9.
4.3.2 Outline of Waves Statistics
Let us assume that a wave record of sea surface oscillation at a given location
is available (see, for example, Fig. 3.10). If the sea surface is so irregular, how
can we determine the probability of the occurrence of waves with a particular
height, say 1 m or 3 m above mean sea level? What is the most frequently
observed surface displacement? To answer these and other questions we have
to know the probability density distribution of the surface displacement. In
most cases, the probability density distribution of the ordinates of ocean surface
waves takes the form of the so called normal distribution or Gaussian function
(Massel, 1996a):
1
[ (( - ()2]
f(() = -[2;0"( exp - 20"~
,
(4.80)
in which ( is the mean value of surface displacement; usually ( = 0 and 0"( is
a standard deviation of the distribution defined as:
(4.81 )
in which N is a total number of sea surface ordinates taken into account in
the summation. In oceanographic practice it is common to take records of
about 20 minutes duration, sampled every 1/4 or 1/2 s, depending on the
instrument used. Therefore, the value of N in (4.81) varies between 2 400
and 4 800 readings for 1/2 sand 1/4 s sampling, respectively. More details
on measurement techniques and data collection procedures can be found in
Chap. 9.
The function (4.80) is illustrated in Fig. 4.24, for a wave record with 0"( = 0.5
m and ( = O. The probability density distribution is a symmetrical curve with
maximum at ( = ( = 0 which satisfies the following condition:
1: f(()d( = 1.
(4.82)
This condition expresses the fact that the probability of occurrence of all cases
should be equal to 1.
143
Frequency analysis mainly deals with an evaluation of the distribution of wave
energy among various frequencies and directions. There are two main methods for the development of the frequency spectrum. The traditional method
is based on the Fourier Transform of the correlation function while in the second method, the time series is simply transferred into its Fourier components.
This method is known as Fast Fourier Transform (FFT). We examine these
techniques in more detail in Chap. 9.
4.3.2 Outline of Waves Statistics
Let us assume that a wave record of sea surface oscillation at a given location
is available (see, for example, Fig. 3.10). If the sea surface is so irregular, how
can we determine the probability of the occurrence of waves with a particular
height, say 1 m or 3 m above mean sea level? What is the most frequently
observed surface displacement? To answer these and other questions we have
to know the probability density distribution of the surface displacement. In
most cases, the probability density distribution of the ordinates of ocean surface
waves takes the form of the so called normal distribution or Gaussian function
(Massel, 1996a):
1
[ (( - ()2]
f(() = -[2;0"( exp - 20"~
,
(4.80)
in which ( is the mean value of surface displacement; usually ( = 0 and 0"( is
a standard deviation of the distribution defined as:
(4.81 )
in which N is a total number of sea surface ordinates taken into account in
the summation. In oceanographic practice it is common to take records of
about 20 minutes duration, sampled every 1/4 or 1/2 s, depending on the
instrument used. Therefore, the value of N in (4.81) varies between 2 400
and 4 800 readings for 1/2 sand 1/4 s sampling, respectively. More details
on measurement techniques and data collection procedures can be found in
Chap. 9.
The function (4.80) is illustrated in Fig. 4.24, for a wave record with 0"( = 0.5
m and ( = O. The probability density distribution is a symmetrical curve with
maximum at ( = ( = 0 which satisfies the following condition:
1: f(()d( = 1.
(4.82)
This condition expresses the fact that the probability of occurrence of all cases
should be equal to 1.