underlying approach to using spectral representations in engineering, discuss the basic approach for the

simplified spectral approaches, and describe how the spectral information can be used. However, the

underlying statistical theory and assumptions will only be touched upon and details of the derivations will

only be referenced.

(b) The easiest place to begin is with a nonrigorous discussion of what a spectral analysis of a single-

point measurement of the surface can produce and then generalize it to the case of a sea surface. The

following sections would then describe of the procedure.

(c) Considering a single-point time-history of surface elevation such as in Figures II-1-25, II-1-31, and

II-1-32, spectral analysis proceeds from viewing the record as the variation of the surface from the mean and

recognizes that this variation consists of several periodicities. In contrast to the wave-by-wave approach,

which seeks to define individual waves, the spectral analysis seeks to describe the distribution of the variance

with respect to the frequency of the signal. By convention, the distribution of the variance with frequency

is written as *E(f) *or *S(f) *with the underlying assumption that the function is continuous in frequency space.

The reason for this assumption is that all observations are discretely sampled in time, and thus, the analysis

should produce estimates as discrete frequencies which are then statistically smoothed to estimate a

continuum. Although *E(f) *is actually a measurement of variance, it is often called the *one-dimensional *or

estimated by multiplying *E(f) *by *ρg*.

(d) Figures II-1-31 (a regular wave) and II-1-32 (an irregular wave) provide two wave records and their

spectrum. One immediate value of the spectral approach is that it tells the engineer what frequencies have

significant energy content and thus acts somewhat analogous to the height-period diagram. The primary

disadvantage of spectral analysis is that information on individual waves is lost. If a specific record is

analyzed, it is possible to retain information about the phases of the record (derived by the analysis), which

allows reconstruction of waves. But this is not routinely done.

(e) The surface can be envisioned not as individual waves but as a three-dimensional surface, which

represents a displacement from the mean and the variance to be periodic in time and space. The simplest

spectral representation is to consider *E(f,θ)*, which represents how the variance is distributed in frequency *f*

and direction *θ *(Figure II-1-33). *E(f,θ) *is called the 2-D or directional energy spectrum because it can be

multiplied by *ρg *to obtain wave energy. The advantage of this representation is that it tells the engineer about

the direction in which the wave energy is moving. A directional spectrum is displayed in Figure II-1-34 with

its frequency and direction spectrums.

(f) The power of spectral analysis of waves comes from three major factors. First, the approach is

easily implemented on a microchip and packaged with the gauging instrument. Second, the principal

successful theories for describing wave generation by the wind and for modelling the evolution of naturalsea

states in coastal regions are based on spectral theory. Third, it is currently the only widely used approach for

measuring wave direction. A final factor is that Fourier or spectral analysis of wave-like phenomena has an

enormous technical literature and statistical basis that can be readily drawn upon.

II-1-78

Water Wave Mechanics

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