(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 *T*r 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 *T*r is dependent on *N*, *M*,

and *∆t *as *T*r = 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 ω*t*

(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 '

(II-1-146)

2

m&4

m0

4

4

'2

'

2

(i) Thus the quantity *a*2/2 represents the contribution to the variance *σ*2 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 E*2 and a *one-sided spectrum E*1 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 *E*2 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 *E*1 (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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