Ogitvie 



simple task for a computer to figure out the velocity field and to pro- 

 duce streamlines. Figure (3-i) shows the streamlines around a 

 Series 60 hull, calculated from the near-field slender -body solution 

 by Tuck and Von Kerczek [ 1968] . The upper boundary of the figure 

 is the rigid-wall streamline. Figure (3-2) shows the same stream- 

 lines in two other views. These drawings are accurate (in principle) 

 to order c. This means, loosely speaking, that they show the 

 streamlines on a scale which is appropriate for measuring beann 

 and draft of the ship. Thus, we see that some of the streamlines 

 start near mid- draft, pass under the bottonn, then return to approxi- 

 mately their original depth. These are variations which show on a 

 scale intended for measuring quantities which are 0(c). 



The wave height, on the contrary, is O(c^) , as we found 

 earlier. Therefore it should not show in these figures. Our 

 assumptions have led to the conclusion that wave height is small 

 compared with beam and draft. Thin- ship theory, on the other hand, 

 predicts that wave height and beam are comparable -- without being 

 very explicit about the ratio of wave height to draft. 



In the section of Fig. (3-2) showing hydrodynamic pressure 

 along streamlines , only the waterplane curve (denoted by W) is really 

 consistent. On any streamline, the pressure will vary mostly because 

 of the changing hydrostatic head along the streamline. Such pressure 

 variations are 0(e). If we were to work out a second-order theory 

 and plot the streamlines , the shift in streamline position from first- 

 order theory to second-order theory would lead to a hydrostatic pres- 

 sure change which is 0(€^). This is the same as the order of magni- 

 tude of the hydrodynamic pressure, but it is ignored in the figure. 



On the other hand, if we were inside the ship measuring 

 pressure at a point on the hull, we would not care which streamline 

 went past that point. We could use the Bernoulli equation to esti- 

 mate the pressure at any point, and the estimate consistent to order 

 €^ would be found from the equation: 



= £ +gz + U$, + i (<i>f + ^f ). 



3.2. The High-Speed, Steady- Motion Problem 



In the preceding analysis, we have said nothing explicit 

 about the speed other than assuming that it was finite. The first 

 term in the velocity- potential expansions was Ux, and all other 

 terms were assumed to be small in comparison. 



In principle, there is no reason to provide or allow a con- 

 nection between Froude number and our slenderness expansion 

 parameter. However, the practical manner in which a perturbation 

 analysis is used may justify our making such an unnatural assumption. 

 In practice, we work out an asymptotic expansion, which provides 



738 



