Pien 



forebody does not generate bow waves. This approach is suggested by Inui's 

 work. We may recall that Inui found that bow wave amplitudes are close to 

 those predicted by theory, particularly if they are small, but that stern wave 

 amplitudes are significantly smaller than those predicted. The ratio of observed 

 to predicted stern wave amplitude is Inui's parameter /3, and it becomes quite 

 small at low Froude numbers. At the Froude numbers around 0.3, for which 

 Dr. Pien has designed his models, the value of /3 for Inui's Model S-201 is under 

 0.7, and for S-202 is even smaller, just above 0.5. This means that the stern 

 waves, which by theory for these double-ended hulls should be as big as the bow 

 waves, are in reality only a little over half as big and so cannot do much to can- 

 cel the bow waves. Worse, they are generated in the frictional wake which 

 moves in the same direction as the ship, and so their transverse components 

 must have a shorter wave length than the transverse components of the bow 

 wave if the stern wave pattern is to move with the ship. Waves must both move 

 in the same direction and have the same wave length if they are to cancel. It is 

 therefore evident that in practice we cannot expect cancellation of the trans- 

 verse portions of the bow wave by the transverse portions of the stern wave 

 even to the extent suggested by the existence of non-zero values of /3 . It is 

 possible to show, also, that transverse components of the bow wave should not 

 penetrate into the wake at all, but should be reflected from its boundary so that 

 cancellation becomes impossible. This, of course, can also be deduced from 

 Inui's experimental observation of the wave-shadow effect. Because of this we 

 should not expect complete cancellation of bow waves by stern waves even at 

 Froude numbers much higher than 0.3 where the value of /3 approaches unity 

 much more closely. 



Actually, it is possible not only to provide an explanation for Inui's semi- 

 empirical parameter /3 from the fact that viscosity causes water to be dragged 

 along with the ship, but an estimate of how much this reduces the velocity of 

 water relative to the stern. As a result of viscosity the stern waves are gener- 

 ated by water moving at a velocity relative to the hull which is somewhat less 

 than that of the water which generates the bow waves. How much less can be 

 deduced by working backwards from Inui's results. This has been done in Fig. 

 1, where the ratio c^/c is that ratio of relative speed of ship and water at the 

 stern to the forward speed of the ship which is required to fit Inui's curves of (3 

 vs Froude number. As shown in Fig. 1, the ratio does not change much over a 

 wide range of speeds. Although the bow, as Pien points out, should be optimized 

 at a speed close to the speed of the ship (he optimized at F = 0.28 for a ship 

 with actual speed F = 0.32), it follows from the data shown in Fig. 1 that the 

 stern should be optimized for a much lower speed. For example, referring to 

 the figure, if we were to optimize the stern of hull S-202 to operate at a ship 

 speed of F = 0.32, we should optimize the stern at a Froude number {c^/c) 

 (0.32) = (0.66) (0.32) - 0.21. 



A second point which Dr. Pien makes is that the conventional hull form pa- 

 rameters must be disregarded when hull forms are optimimized. Certainly I 

 find this to be true. I have just finished a calculation using the method of steep 

 descent to decrease the wave resistance of a destroyer ty^e ship intended to 

 operate at 30 knots, and in the calculation I held constant the sectional area 

 curve as well as the load waterline, the sound dome, and the draft. The calcu- 

 lation, which started with a hull designed by a good naval architect, had to make 



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