Y - 0, F - 0.S 



Fig. 9a — Real and Imaginary Parts of the Functions 



K|^(t) and Ko(t) for a Simple Ship Bow Form with 

 P = 12° (Top) and 3° (Bottom), y = and F = 0.5 



and that this thin-ship limit is not uniformly valid in the 

 limit t -* ". More generally, the limiting processes /3q -* 

 and t -* 00 cannot be interchanged. 



Figures 9a,b depict the real and/or imaginary parts 

 of the functions {l + i^)^^^K^(t) and (l+i^)^^\{l) for the 

 simple bow shape defined by equations (40a,b), where the 

 nondimensional draft d and the maximum flare angle y 

 are taken equal to d = 0.1 and y = 0, and four values 

 of the half-entrance-angle p are considered, namely p = 

 12°, 3°, 1° and 20'. The Froude number based on the 

 length of the bow region is taken equal to F = 0.5 in the 

 numerical results presented in figures 9a,b and in figures 

 10 and 11 considered further on. The values of 1//3q = 

 l/tan/l corresponding to the values of p equal to 12°, 3°, 

 1° and 20' are approximately equal to 4.7, 19, 57 and 

 172, respectively. The functions (1 -l-t^)'^^K|^(t) and 

 (1 -(-t^)'^^KQ(t) are depicted for < t < 10 in figures 9a,b. 

 Differences between the approximations K,^ and Kq, 

 especially their imaginary parts, can be seen to be 

 substantial in figure 9a corresponding to |3= 12° and 3°. 

 Figure 9b shows that differences between the imaginary 

 parts of the functions K^^ and Kq remain appreciable even 



-0.00S 



Y-0,F>0.5 



Fig. 9b — Real and Imaginary Parts of the Functions 



K|^(t) and KQ(t) for a Simple Ship Bow Form with 

 /3 = r (Top) and 20' (Bottom), y = and F = 0.5 



for values of p equal to 1° and 20', which are quite 

 small, and for values of t that are much smaller than 

 l//3o. 



The top and bottom parts of figure 10 depict the 

 real and imaginary parts of the functions (1 -(■t^)'^^K|^(t) 

 and (1 -l-t^)''^^K(,(t), respectively, for the previously- 

 considered simple bow shape with d = 0.1, |J = 12°, F 

 = 0.5 and for two values of the fiare angle, namely for y 

 = and 45°. The top part of the figure shows that 

 differences between the curves corresponding to y = 

 and y = 45° are faily small, and are appreciable only for 

 small values of t, for the Michell thin-ship approximation 

 K|^. In particular, the asymptotic approximation 

 ^^(t) ~ 4/3(, sin(v^/2) exp( - iv^t/2)/t^ as t - oo, 

 which may be obtained from equations (43) and (44), is 

 independent of y. The bottom part of figure 10 shows 

 differences between the curves corresponding to y = 

 and y = 45° for the slender-ship approximation Kg that 

 are significantly larger than those for the Michell 

 approximation K^, especially for intermediate values of t 

 in the vicinity of t = l/tan/3 ~ 4.7. The fiare angle y 

 thus has a pronounced effect upon the behavior of the 



11 



