How the bulge wave works. The water wave is travelling to the right. The bulge leads the wave by 90°. Open arrows show the particle velocity in the bulge: forward when the pressure is high, backward when it is low, just as in the water wave. Black arrows show the movement of the tube wall. It is moving inwards when the wave pressure is high; so the wave does work on the tube. When the wave pressure is low, it is sucking the tube outwards, again doing work. So energy is transferred from the sea wave to the bulge wave.
If the tube snakes with the waves (travelling to the right), gravity drives the bulge forward along the tube. But the velocities match; so the wave keeps up with the bulge. In effect, the bulge is surfing in front of the wave, picking up energy and growing larger as it runs. Another point of view: in front of the wave the water is rising and lifts the bulge putting in energy; behind the wave the water is falling; so the tube is dropping losing energy, but at the waist there is less water to drop. (Online version in colour.)
Wall tension T versus radius r for an all rubber tube. The required tension (line 2) intersects the rubber characteristic (line 1) at equilibrium point A. As pressure p increases the equilibrium moves up to B, then jumps to C, creating the aneurism.
Wall tension T versus rubber elongation x if tube circumference comprises a metres of fabric for each metre of unstretched rubber. The origin of line 2 is displaced to the left. Intersection point A of lines 1 and 2 becomes stable, no aneurism.
Bulge speed versus static pressure for a tube with 23% rubber. Line is theoretical calculation using parameters given in the box and a stress–strain curve for the rubber; triangles are measurements (made by S. Rimmer, Checkmate Seaenergy Ltd) in the tank.
CW versus wave period for tune 1.8 s. SSU4 tank test (circles); simulation TD13 (squares); simulation TD14 with loss angle 2° (triangles); simple theory (equations (4.6) and (5.3)) (solid line). (Online version in colour.)
CW for 8 s waves, tubes of various lengths and diameters. Simulation results for a Pierson–Moskowitz spectrum with peak frequency at 0.125 Hz, power level 50 kW m−1. The length of the tube is along the horizontal axis. The open points (circles, diameter 2.5 m; diamonds, diameter 5 m; triangles, diameter 8 m) are for tubes with a matched load at the stern. The solid points (circles, diameter 2.5 m; triangles, diameter 4 m) are for the distributed PTO with 11 pairs of one-way valves equally spaced along the length. Because the waves are random the CW has some scatter.
Fatigue life of rubber adapted from Cadwell et al. , Fig. 8. L is the length of the sample of initial length L0. Life is plotted against minimum strain for various ranges of the strain excursion ΔL.