## Where the Power Comes From

In common with wind, wave power is difficult to exploit because of its diffuseness and its intermittency. But in addition, to extract energy from the waves it is necessary to go where the waves are powerful, which is also where the seas or oceans are deep and very inhospitable. Wave power can be thought of as concentrated solar power, formed when winds generated by differential heating of the atmosphere sweep over open expanses of sea or ocean transferring some of their energy into water waves. The amount of energy transferred and hence the magnitude of the resulting waves depends on the wind speed, the length of time the wind blows and the expanse of water surface over which it blows (termed the 'fetch'). In this way the original solar power levels ~ 1000 W/m2 can be translated into ocean waves exhibiting power levels of the order of 100 MW/km of wave front as we shall see.

The nature of ocean waves is becoming increasingly well understood from studying water movement in special tanks and with the aid of sophisticated computer modelling [15, 16]. The complex motions of the water occur not just in the visible surface waves, but also well below the surface. In fact, the presence of the wave at the surface is reflected in water movements down to a depth which is of the order of about a half wavelength of the surface wave. In the deep ocean, the wave length of the surface waves, that is the crest-to-crest distance, is approximately equal to gravitational acceleration divided by the product of twice n (3.412) and frequency squared [17]. An ocean swell exhibits a typical frequency of 0.1 Hz which gives a wavelength of 156 m. Hence the depth below which wave action is not discernable in ocean waters is of the order of 80 m. This knowledge is important to the design of effective wave energy collectors. The velocity at which the wave travels (v in m/s) is given approximately by v = 1.25 VX, where wavelength A is measured in metres. Typically the crest velocity of a deep ocean wave is 16 m/s. However, the velocity expression also tells us that waves of different wavelengths travel at different speeds. The fastest waves in a storm are the ones with the longest wavelength. Observant sea watchers may have noted that when waves arrive on the coast after a storm far out to sea, the first ones to arrive are the long wavelength swells - not a bit like a high class social event!

When several wave trains are present, as is always the case in the ocean, the waves form groups which appear as higher than average ridges or pulses of water. In deep water the groups travel at a velocity (termed the group velocity) that is half of the phase speed [18]. Group velocity is associated with the energy transmission and is important in determining the power of the waves. In a wave tank it is feasible to follow a single wave in a pulse. When one does, it is possible to see the wave appearing at the back of the group, growing and finally disappearing at the front of the group. As the water depth decreases towards the coast, this will have an effect on the speed of the crest and the trough of the wave; the crest begins to move faster than the trough. This causes a phenomenon with which everyone is familiar, namely surf and breaking waves.

The power of ocean waves can be captured by devices that oscillate in response to the wave motion. The available power per unit width of regular sinusoidal waves depends on the water density p, gravitational acceleration g, the mean wave height H and the wavelength. It is given by a simple formula derived from energy considerations, which can be found in most textbooks dealing with water waves [16]. For strong 10 ft peak to trough, ocean swells oscillating with a frequency of typically 0.1 Hz the crude formula suggests available powers in the range of 100 kW/m (100 MW/km). Measurements from a mid-Atlantic weather station indicate that average wave power levels of 80 MW/km of wave front are potentially available there, if wave machines could be located safely and reliably, in hostile deep ocean environments. A very 'tall order' as we shall see. Nearer shore, but still in deep water, such as to the west of the Outer Hebrides of Scotland, the wave power is somewhat lower at 50 MW/km. However, it is certainly more than enough to merit examination as an exploitable resource.