
Refraction Focusing—"DAMATOLL"

In 1946 Robert S. Arthur presented a paper in which he desribed the refraction of water waves by idealized islands having circular bottom contours. By the proper choice of bottom profile, Arthur (1946) showed that the power from that portion of the wave front directly influenced by the island is focused on the center of the island, as illustrated in Figure 4.39a. He studied the focusing phenomenon by analyzing the changes in the directions of the wave orthogonal or rays as the wave refracts on the island.
Using the results of the Arthur (1946) study, Wirt (1976) of the Lockheed California Company showed that the most suitable island profile for wave energy conversion is the one that yields logarithmic spiral orthogonals, that is, those described by

r = r_{0}exp[ ( &theta  &theta_{0} ) cot ( &theta_{0} ) ] (4.124)

where r_{0} is the radius of the first shoaling contour and &theta_{0} the angle of the nonrefracting orthogonal. Referring to Figure 4.39b, the profile of the island is given by the equation

d_{1} = λ_{0} ∕ 2_{&pi} ( r ∕ r_{0} )tanh^{1}( r ∕ r_{0}) (4.125)

The waves refract in the pattern shown in Figure 4.39_{a} until they break. The broken waves then become surges and are guided tinto a vertical shaft by vanes, as illustrated in Figure 4.40. The entrance of the guide vanes are positioned just before the breaking point of the waves. The vanes are designed to give the surge a tangential velocity as the water enters the veritcal shaft, as illustrated in Figure 4.40. The water then descends as a water column with a rotational motion and thus acts as a fluid flywheel. At the bottom of the shaft is a vertical axis turbine, which is illustrated in the crosssectional sketch in Figure 4.41. The turbine, in turn, drives an electrical generator. After [exiting] this turbine, the water then passes through a radial diffuser and returns to the ocean.
The system described by Wirt (1976), called the "DAMATOLL," is designed to adjust its vertical position as the mean water level changes with the tides. This is accomplished by giving the system a relatively small amount of positive boyancy and by using an adjustable mooring, as illustrated in Figure 4.41. The island, or atoll, can be compliant walled and water filled. This design feature allows for ease in both transportation to the site and positioning. After the system is placed in the water, it floats as a result of flotation chambers located in the rigid central portion of the system. The flexible chambered walls of the atoll are then filled with water tht is pumped in at a high pressure to maintain the desired profile [given by the equation (4.125)]. These design features should keep the costs of the DAMATOLL system to a minimum. A sketch of an operational system is presented in Figure 4.42.

h > &lambda_{0} ∕ 2 = 31.3 ft ( 9.55 m )

Therefore the incident waves are deep water waves. The atoll is assumed to have a diameter of 200 ft (61.0 m) at its first refracting contour; thus

r_{0} = 200 ft (61.0 m)
d_{0} = 31.3 ft (9.55 m)

The wave power available to the DAMATOLL is obtained from equation (4.112), where b in that equation [is] replaced by 2r_{0}. Thus in deep water, where the group velocity is

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