EM 1110-2-1100 (Part II)
30 Apr 02
(f) There are numerous intricacies involved in the application of these discrete formulas, ranging from
the length of time series necessary to digitizing frequency and many others. For unfamiliar users, most
computer library systems now have FFT (Finite Fourier Transform) algorithms available to perform the
above computations. Part VII-3 of the CEM provides a discussion of the methods. Some general guidelines
are provided next.
(g) In actual practice, the total data length is divided into M smaller segments with equal number of data
points N. By letting N be a power of 2 for computational efficiency, the result then is averaged over the M
sections. In an FFT, the variables M, N, and ∆t have to be independently selected, though Tr and ∆t are fixed
for a given record so that the total number of data points can be obtained from these values. Therefore, the
only choice that has to be made is the number of sections M. Traditionally, the most common values of N
used range from 512 to 2,048, while the value of M is usually 8 or greater. Since Tr is dependent on N, M,
and ∆t as Tr = M N ∆t, then higher N and M values in general yield better resolution and high confidence in
the estimate of spectra. The larger the N, the more spiky or irregular the spectrum, and the smaller the N, the
smoother the spectrum (Cooley and Tukey 1965; Chakrabarti 1987).
(h) To better understand the wave spectrum by the FFT method, consider first the wave surface profile
of a single-amplitude and frequency wave given by a sinusoidal function as
η(t) ' a sin ωt
(II-1-145)
where a and ω are the amplitude and frequency of the sine wave. The variance of this wave over the wave
period of 2π is
1
2π m0
2π
σ2 ' [η(t)]2 '
a 2 sin2 2πft d(2πft)
(II-1-146)
2
a
m&4
m0
4
4
E 1(f) df '
E 2(f) df
'2
'
2
(i) Thus the quantity a2/2 represents the contribution to the variance σ2 associated with the component
frequency ω = 2πf (Figure II-1-35). The connection between the variance, wave energy, and the wave energy
spectrum is now more obvious since these all are proportional to the wave amplitude (or height) squared. For
consistency of units, an equality between these quantities requires that the wave spectrum not include the ρg
term.
(j) The difference between a two-sided spectrum E2 and a one-sided spectrum E1 as illustrated in Figure
II-1-36 is quite important. Note that the two-sided spectrum is symmetric about the origin, covering both
negative and positive frequencies to account for all wave energy from -4 to +4. But, it is customary in ocean
engineering to present the spectrum as a one-sided spectrum. This requires that the spectral density ordinates
of E2 be doubled in value if only the positive frequencies are considered. This is the reason for introducing
a factor of two in Equation II-1-146. This definition will be used subsequently throughout Part II-1; thus,
it is henceforth understood that E(f) refers to E1 (Figures II-1-35 and II-1-36).
(k) By an intuitive extension of this simple wave, the variance of a random signal with zero mean may
be considered to be made up of contributions with all possible frequencies. For a random signal using the
above equations, we find
Water Wave Mechanics
II-1-83
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