The instabilities first appeared in areas where the body curvature was large. 

 No completely satisfactory explanation of the problem was found, but numerical 

 experimentation led to the use of underrelaxation factors between 0.75 and 1.0 

 on the body. Such underrelaxation factors on the body eliminated the 

 instabilities. 



Figures 7-10 depict some results of computing the free-surface position 

 during the first two periods of forced motion with the amplitude such that 

 hQ/b = 0.4 and the frequency such that bo /g =2. A time step of At = 0.02 

 resolved one period of motion into about 314 time steps. Figures 7 and 8 show 

 the free surface and the body with a fixed coordinate system. Figure 7, 

 corresponding to times between 5.4 and 8.8, shows a rising body and a sinking 

 free surface near the body; Figure 8, for times between 9.0 and 11.4, shows a 

 descending body and a rising free surface. Figures 9 and 10 show similar 

 results, but the coordinate system in these figures is fixed to the heaving 

 body. The time-dependent details of the free surface near the body are 

 clearer in this frame of reference. Figure 9 shows a rising body between the 

 times 6.2 and 9.0; Figure 10, a descending body between the times 9.2 and 

 12.2. It is interesting to note that the slope of the free surface near the 

 body is largest at about the time the body has attained its maximum height. 

 Results for computing the vertical force on the heaving U-shaped body are 

 depicted in Figure 11. From the figure it is seen that the force has become 

 periodic in less than one period of forced motion. Also shown on the figure 

 is a curve of the force versus time predicted from the second-order theory of 

 Papanikolaou and Nowacki [18]. The other curve is a linear magnification of 

 the results from calculating the force when the amplitude of motion is eight 

 times smaller (hg/b = 0.05) but the frequency of the motion is the same. 



20 



