16.3 Waves and the Concept of a Wave Spectrum
If we look out to sea, we notice that waves on the sea surface are not simple sinusoids. The surface appears to be composed of random waves of various lengths and periods. How can we describe this surface? The simple answer is, Not very easily. We can however, with some simplifications, come close to describing the surface. The simplifications lead to the concept of the spectrum of ocean waves. The spectrum gives the distribution of wave energy among different wave frequencies of wave-lengths on the sea surface.
The concept of a spectrum is based on work by Joseph Fourier (1768 - 1830), who showed that almost any function ζ (t) (or ζ (x) if you like), can be represented over the interval -T/2 < t < T/2 as the sum of an infinite series of sine and cosine functions with harmonic wave frequencies:
f = 1/T is the fundamental frequency, and nf are harmonics of the fundamental frequency. This form of ζ (t) is called a Fourier series (Bracewell, 1986: 204; Whittaker and Watson, 1963: §9.1). Notice that a0 is the mean value of ζ (t) over the interval.
Equations (16.18 and 16.19) can be simplified using
where i = square root of (-1). Equations (16.18 and 16.19) then become:
Zn is called the Fourier transform of ζ (t).
The spectrum S(f) of ζ (t) is:
where Z* is the complex conjugate of Z. We will use these forms for the Fourier series and spectra when we describing the computation of ocean wave spectra.
We can expand the idea of a Fourier series to include series that represent surfaces ζ (x, y) using similar techniques. Thus, any surface can be represented as an infinite series of sine and cosine functions oriented in all possible directions.
Now, let's apply these ideas to the sea surface. Suppose for a moment that the sea surface were frozen in time. Using the Fourier expansion, the frozen surface can be represented as an infinite series of sine and cosine functions of different wave numbers oriented in all possible directions. If we unfreeze the surface and let it evolve in time, we can represent the sea surface as an infinite series of sine and cosine functions of different wave-lengths moving in all directions. Because wave-lengths and wave frequencies are related through the dispersion relation, we can also represent the sea surface as an infinite sum of sine and cosine functions of different frequencies moving in all directions.
Note in our discussion of Fourier series that we assume the coefficients (an, bn, Zn) are constant. For times of perhaps an hour, and distances of perhaps tens of kilometers, the waves on the sea surface are sufficiently fixed that the assumption is true. Furthermore, non-linear interactions among waves are very weak. Therefore, we can represent a local sea surface by a linear super-position of real, sine waves having many different wave-lengths or frequencies and different phases traveling in many different directions. The Fourier series in not just a convenient mathematical expression, it states that the sea surface is really, truly composed of sine waves, each one propagating according to the equations we wrote down in §16.1.
The concept of the sea surface being composed of independent waves can be carried further. Suppose I throw a rock into a calm ocean, and it makes a big splash. According to Fourier, the splash can be represented as a superposition of cosine waves all of nearly zero phase so the waves add up to a big splash at the origin. Furthermore, each individual Fourier wave then begins to travel away from the splash. The longest waves travel fastest, and eventually, far from the splash, the sea consists of a dispersed train of waves with the longest waves further from the splash and the shortest waves closest. This is exactly what we see in Figure 16.1. The storm makes the splash, and the waves disperse as seen in the figure.
Sampling the Sea Surface
Working with a trace of wave-height on say a piece of paper is difficult, so let's digitize the output of the wave staff to obtain
Notice that ζj is not the same as ζ (t). We have absolutely no information about the height of the sea surface between samples. Thus we have converted from an infinite set of numbers which describes ζ (t) to a finite set of numbers which describe ζj. By converting from a continuous function to a digitized function, we have given up an infinite amount of information about the surface.
The sampling interval Δ defines a Nyquist critical frequency (Press et al., 1992: 494)
Figure 16.4 illustrates the aliasing problem. Notice how a high frequency signal is aliased into a lower frequency if the higher frequency is above the critical frequency. Fortunately, we can can easily avoid the problem: (i) use instruments that do not respond to very short, high frequency waves if we are interested in the bigger waves; and (ii) chose Dt small enough that we lose little useful information. In the example shown in Figure 16.3, there are no waves in the signal to be digitized with frequencies higher than Ny = 1.5625 Hz.
Let's summarize. Digitized signals from a wave staff cannot be used to study waves with frequencies above the Nyquist critical frequency. Nor can the signal be used to study waves with frequencies less than the fundamental frequency determined by the duration T of the wave record. The digitized wave record contains information about waves in the frequency range:
where T = NΔ is the length of the time series, and f is the frequency in Hertz.
Calculating The Wave Spectrum
for j = 0, 1, · · · , N - 1; n = 0, 1, · · · , N - 1. These equations can be summed very quickly using the Fast Fourier Transform, especially if N is a power of 2 (Cooley, Lewis, and Welch, 1970; Press et al., 1992: 542).
The simple spectrum Sn of ζ, which is called the periodogram, is:
where SN is normalized such that:
thus the variance of ζj is the sum of the (N/2 + 1) terms in the periodogram. Note, the terms of Sn above the frequency (N/2) are symmetric about that frequency. Figure 16.5 shows the periodogram of the time series shown in Figure 16.2.
The periodogram is a very noisy function. The variance of each point is equal to the expected value at the point. By averaging together 10-30 periodograms we can reduce the uncertainty in the value at each frequency. The averaged periodogram is called the spectrum of the wave-height (Figure 16.6). It gives the distribution of the variance of sea-surface height at the wave staff as a function of frequency. Because wave energy is proportional to the variance (16.12) the spectrum is called the energy spectrum or the wave-height spectrum. Typically three hours of wave staff data are used to compute a spectrum of wave-height.
This outline of the calculation of a spectrum ignores many details. For more complete information see, for example, Percival and Walden (1993), Press et al., (1992: §12), Oppenheim and Schafer (1975), or other texts on digital signal processing.
|Department of Oceanography, Texas A&M University
Robert H. Stewart, email@example.com
All contents copyright © 2005 Robert H. Stewart,
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Updated on November 15, 2006