6.8: Characteristics of Real Ocean Waves

The origin of almost all ocean waves is wind. Tides and tsunamis also count as waves, but of course at different frequencies. Sustained wind builds waves that are bigger in amplitude and longer in wavelength - hence their frequency decreases. Waves need sufficient physical space, called the fetch, to fully develop. When the wind stops (or the wave moves out of a windy area), the amplitude slowly decays, with characteristic time \(\tau = g^2 / 2 \nu \omega^4\). This rule says that low-frequency waves last for a very long time!

The spectra of ocean waves are reasonably modeled by the standard forms, including JONSWAP, Pierson-Moskowitz, Ochi, and Bretschneider; these have different assumptions and different applications. The conditions of building seas and decaying seas (swell) are different; in the former case, the spectrum is quite wide whereas it may be narrow for the latter. Further details can be found in subject texts, including the Principles of Naval Architecture (E.V., Lewis, ed. SNAME, 1989).

Most importantly from a design point of view, it has been observed that extreme events do NOT follow the Rayleigh distribution - they are more common. Such dangers are well documented in data on a broad variety of processes, including weather, ocean waves, and some social systems. In the case of ocean waves, nonlinear effects play a prominent role, but a second factor which has to be considered for long-term calculations is storms. In periods of many years, intense storms are increasingly likely to occur, and these create short-term extreme seas that may not be well characterized at all in the sense of a spectrum. For the purpose of describing such processes, the Weibull distribution affords some freedom in shaping the ”tail.” The Weibull cpf and pdf are respectively:

\begin{align} P(h<h_o) \, &= \, 1 - e^{-(x - \mu)^c / b^c}; \\[4pt] p(h) \, &= \, \dfrac{c(x - \mu)^{c-1}}{b^c} e^{-(x-\mu)^c / b^c} \end{align}

It is the choice of \(c\) as a real number other than two which makes the Weibull a more general case of the Rayleigh distribution. \(b\) is a measure related to the standard deviation, and \(\mu\) is an offset applied to the argument, giving further flexibility in shaping. Clearly \(x > \mu\) is required if \(c\) is non-integer, and so \(\mu\) takes the role of a lower limit to the argument. No observations of \(h\) below \(\mu\) are accounted for in this description.

Here is a brief example to illustrate. Data from Weather Station India was published in 1964 and 1967 (see Principles of Naval Architecture), giving a list of observed wave heights taken over a long period. The significant wave height in the long-term record is about five meters, and the average period is about ten seconds. But the distribution is decidedly non-Rayleigh, as shown in the right figure below. Several trial Weibull pdf’s are shown, along with an optimal (weighted least-squares) fit in the bold line. The right figure is a zoom of the left, in the tail region.

Figure \(\PageIndex{1}\): compare and contrast the long-term distribution of actual wave heights against the data set's Rayleigh fit and several trial Weibull fits.

Armed with this distribution, we can make the calculation from the cpf that the 100-year wave is approximately 37 meters, or \(7.5 \bar{h}^{1/3}\). This is a very significant amplification, compared to the factor of three predicted using short-term statistics in Section 5.6, and reinforces the importance of observing and modeling accurately real extreme events.