The Fourier spectrum of waves in the Pacific Ocean is analysed. The power in 4 s waves is more regular than the power in 10 s waves and larger than expected.
1. Opening remarks by the conference organizer
Welcome to this Discussion Meeting on ‘The peaks and troughs of wave energy: the dreams and the reality’ organized by Rod Rainey, John Chaplin and myself. People have dreamt of harnessing the waves for centuries, but the attempts have often ended in disaster. Many of the world experts in this field are here today and we thank them for coming. Over these 3 days, we hope to present an up-to-date survey of where we are, the theoretical framework, the main practical difficulties and possible solutions; to share our knowledge and pool our expertise. There is a huge collection of talent in this room. If we work together, surely we can make it happen.
In this spirit, I would like to share with you something I have discovered recently. There is more power in the short waves than we suppose.
2. There is more power in the short waves than we suppose
We are accustomed to wave tables which show the significant wave height Hs and the so-called energy period Te which at any moment identifies the waves with the highest energy. This is usually the long rolling swell. But as any photo of the sea will show, riding on top of the long waves, there are plenty of short waves. The short waves are always there, but they are not reported by the European Marine Energy Centre (EMEC) Ltd, UK (http://www.emec.org.uk). How much energy do they contain? We need the Fourier spectrum, which is not readily available in the UK.
Fortunately, the USA (National Oceanic and Atmospheric Administration, National Data Buoy Center, Stennis Space Center, MS 39529, USA). publishes free of charge the complete Fourier spectrum of the power in the waves at many locations, every half hour throughout the year, in steps of 0.01 Hz. They cover few sites near Europe but the waves in the north Pacific near Alaska may resemble those near Scotland, so I looked at the data for Shumagin (figure 1)  and, as a preliminary exercise, compared the wave power at 10 s period with the power at 4 s, both for the same fractional frequency band df/f=0.28.
The power spectra for the 10 and 4 s waves for the first half of the year 2008 are shown in figure 2. At 10 s (figure 2a) the average power in this frequency band is about 4 kW m−1 with some calm periods and peaks up to 40 kW m−1, sometimes 80 kW m−1. The power level is quite variable. The power at 4 s (figure 2b) averages about 0.4 kW m−1 but is much steadier with few gaps and the high peaks are rare.
The data for July–December 2008, plotted with the same scales, are given in figure 3. In July and August, there is hardly any power at 10 s, but in the winter months there are huge peaks which could destroy your machine. By contrast, the 4 s power is still there in the summer, although down to about half. In the winter, the power is up, but much more regular than the long waves. If you are building a wave power machine, which power graph would you prefer to work with?
The same conclusions follow from the Pierson–Moskowitz spectrum plotted against wave period in figure 4 for various wind speeds. The 10 s waves vary enormously with wind speed. But the 4 s waves are always there and almost fully developed even in light winds; they do not increase much as the wind speed rises because they have reached their limit. On average, the 4 s waves are three times steeper than the 10 s waves. Short steep waves are the best for a wave power converter.
3. Cost of machine per kilowatt collected
We all know from experience that small wave power converters are easier to make and less fragile than their big brothers. The dimensions of the machine scale with the wavelength, so one expects the structural weight to scale as λ3. However, as a device gets larger structural components must be thicker in proportion, so it is reasonable to suppose that the mass of a wave power converter, and its cost, will scale as λ3.5, that is as T7. If the wave height is proportional to wavelength, the power over a frontage proportional to wavelength also scales as T7. As a rough guess, the cost of the device per kilowatt captured should then be almost independent of scale. But we have seen that the 4 s waves at Shumagin, averaged over the year, are three times steeper than the 10 s waves. As a result, sixty 4 s machines will get out the same power as one 10 s machine; and their total cost will be about 10 times less!
This is only a preliminary study. Shumagin may not be typical of the waves around Europe. We need the Fourier spectra at Benbecula. Also there are problems in deploying, connecting and maintaining many small machines, in the place of one large one. If you can think of an economical way to do this, then the shorter waves might provide the cheapest and most reliable power.
One contribution of 18 to a Theo Murphy Meeting Issue ‘The peaks and troughs of wave energy: the dreams and the reality’.
- This journal is © 2011 The Royal Society