EM 1110-2-1100 (Part II)
30 Apr 02
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
frequency energy spectrum because (assuming linear wave theory)
of the wave field may
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.
Water Wave Mechanics