4.3 Statistical and Spectral Properties of Waves
147
The highest probability of occurrence is associated with a wave height slightly
smaller than the mean wave height. If the wave height obeys the Rayleigh
distribution (4.86), then the significant wave height (Hs), defined in Chap. 3,
becomes (Massel, 1996a):
(4.92)
Substituting Eq. (4.90) into (4.92) yields:
Hs ~ 1.6H.
(4.93)
In Table 4.1 the relationships between various characteristic wave heights are
given. For example, the mean height of the highest 100 waves is about 1.67
times higher than the significant height.
4.3.3 Spectral Properties of Wind-Induced Waves
In order to clarify the fundamentals of spectral analysis of time series of random,
irregular ocean surface waves, let us begin with analysis of time series for
0< t < T (see Fig. 4.26a). As is shown in many basic mathematical text books
(for example, Hildebrand, 1965), any irregular function can be represented as
a summation of many sinusoidal curves. The basis for such representation is
the Fourier series, named after Joseph Fourier (1768-1830):
(4.94)
or
1
n=N
f(t) = 2"ao + ]; [an cos (nwt) + bn sin (nwt)].
( 4.95)
The frequencies of these curves are multiples of the basic frequency w = 27r IT.
Therefore, we have curves with frequencies 2w, 3w, ... ,nw. By analogy to
acoustics and music we called them harmonics. Accuracy of the representation
(4.95) depends on the complexity of the function f(t) and the number of harmonics, N. For convergence of the series to the function f (t) it is sufficient
that functions f (t) and f' (t) are continuous except at a finite number of points
in the interval (0, T). Fourier series not only represent the function on the
interval (0, T) but also give the periodic extension of the function f outside
this interval.
It is convenient for further analysis to rewrite Eq. (4.95) in the more compact
form:
1
n=N
f(t) = -aD + L en cos (nwt + En),
2
n=l
( 4.96)
147
The highest probability of occurrence is associated with a wave height slightly
smaller than the mean wave height. If the wave height obeys the Rayleigh
distribution (4.86), then the significant wave height (Hs), defined in Chap. 3,
becomes (Massel, 1996a):
(4.92)
Substituting Eq. (4.90) into (4.92) yields:
Hs ~ 1.6H.
(4.93)
In Table 4.1 the relationships between various characteristic wave heights are
given. For example, the mean height of the highest 100 waves is about 1.67
times higher than the significant height.
4.3.3 Spectral Properties of Wind-Induced Waves
In order to clarify the fundamentals of spectral analysis of time series of random,
irregular ocean surface waves, let us begin with analysis of time series for
0< t < T (see Fig. 4.26a). As is shown in many basic mathematical text books
(for example, Hildebrand, 1965), any irregular function can be represented as
a summation of many sinusoidal curves. The basis for such representation is
the Fourier series, named after Joseph Fourier (1768-1830):
(4.94)
or
1
n=N
f(t) = 2"ao + ]; [an cos (nwt) + bn sin (nwt)].
( 4.95)
The frequencies of these curves are multiples of the basic frequency w = 27r IT.
Therefore, we have curves with frequencies 2w, 3w, ... ,nw. By analogy to
acoustics and music we called them harmonics. Accuracy of the representation
(4.95) depends on the complexity of the function f(t) and the number of harmonics, N. For convergence of the series to the function f (t) it is sufficient
that functions f (t) and f' (t) are continuous except at a finite number of points
in the interval (0, T). Fourier series not only represent the function on the
interval (0, T) but also give the periodic extension of the function f outside
this interval.
It is convenient for further analysis to rewrite Eq. (4.95) in the more compact
form:
1
n=N
f(t) = -aD + L en cos (nwt + En),
2
n=l
( 4.96)