16.4 Ocean-Wave Spectra
Ocean waves are produced by the wind. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In designing ships or offshore structures we wish to know the biggest waves produced by a given wind speed. Suppose the wind blows at 20 m/s for many days over a large area of the North Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?
To obtain a spectrum of a fully developed sea, they used measurements of waves made by accelerometers on British weather ships in the North Atlantic. First, they selected wave data for times when the wind had blown steadily for long times over large areas of the North Atlantic. Then they calculated the wave spectra for various wind speeds, and they found that the spectra were of the form (Figure 16.7):
where ω = 2π f, f is the wave frequency in Hertz, a = 8.1 × 10-3, b = 0.74, ω0 = g/U19.5 and U19.5 is the wind speed at a height of 19.5 m above the sea surface, the height of the anemometers on the weather ships used by Pierson and Moskowitz (1964).
For most air flow over the sea the atmospheric boundary layer has nearly neutral stability, and
assuming a drag coefficient of 1.3 × 10-3.
The frequency of the peak of the Pierson-Moskowitz spectrum is calculated by solving dS/dω = 0 for ωp, to obtain
The speed of waves at the peak is calculated from (16.10), which gives:
Hence waves with frequency ωp travel 14% faster than the wind at a height of 19.5 m or 17% faster than the wind at a height of 10 m. This poses a difficult problem: How can the wind produce waves traveling faster than the wind? We will return to the problem after we discuss the JONSWAP spectrum and the influence of nonlinear interactions among wind-generated waves.
The significant wave-height is calculated from the integral of S(ω) to obtain:
Remembering that H1/3 = 4 <ζ 2>1/2, the significant wave-height calculated from the Pierson-Moskowitz spectrum is:
Figure 16.8 gives significant wave-heights and periods calculated from the Pierson-Moskowitz spectrum.
Wave data collected during the JONSWAP experiment were used to determine the values for the constants in (16.36):
where F is the distance from a lee shore, called the fetch, or the distance over which the wind blows with constant velocity.
The energy of the waves increases with fetch:
where x is fetch.
The JONSWAP spectrum is similar to the Pierson-Moskowitz spectrum except that waves continues to grow with distance (or time) as specified by the α term, and the peak in the spectrum is more pronounced, as specified by the γ term. The latter turns out to be particularly important because it leads to enhanced non-linear interactions and a spectrum that changes in time according to the theory of Hasselmann (1966).
Generation of Waves by Wind
|Department of Oceanography, Texas A&M University
Robert H. Stewart, firstname.lastname@example.org
All contents copyright © 2005 Robert H. Stewart,
All rights reserved
Updated on November 15, 2006