0-S larS «■■» 



0» 1«k5 



Fig. 16 — Steepness of the Transverse ( - ) and Divergent 

 ( + ) Waves in the Kelvin Wake for a Simple Ship Bow 



Form, with j3 = 12°, y = 45° and F = 0.5, in the 



Michell Thin-Ship Approximation (Left) and the Zeroth- 



Order Slender-Ship Approximation (Right) 



t-ir,y-*e- 



Km 



KO 



0.2 1/2r^ 0.2 1/21^ 



Fig. 17 — Upper Bound for the Steepness of the 



Divergent Waves in the Kelvin Wake for a Simple Ship 



Bow Form, with /3 = 12° and y = 45°, in the Michell 



Thin-Ship Approximation (Left) and the Zeroth-Order 



Slender-Ship Approximation (Right) 



equations (45) and (67b). Figure 16 also shows a sharp 

 peak in the steepness of the divergent waves at the value 

 of a equal to tan/5/(2 + lan^j)). 



The steepness s _^ (o) of the divergent waves is given 

 by equations (32) and (33). An upper bound for the 

 functions s^(a) may be obtained by using an upper 

 bound for the function |K(t^)| in equation (32). Equation 

 (45) yields the following upper bound for the function 

 |K|^(t)| corresponding to the Michell thin-ship 

 approximation: |K|^(t)| < '^(/i^ as t -♦ <». An upper 

 bound for the function |KQ(t)| is given by equation (66), 

 where the upper bound defined by equation (68) is used 

 for the term |N|. These upper bounds for the functions 



|K^(t)| and |KQ(t)| can be expressed in terms of a by 

 using equation (12). The corresponding upper bounds for 

 the steepness function s^(o) are depicted in figure 17. 

 Comparison of figures 16 and 17 shows that the upper 

 bound for the steepness of the divergent waves in the 

 Kelvin wake depicted in figure 17 is satisfactory for all 

 values of a for the Michell approximation K^, whereas 

 that corresponding to the slender-ship approximation Kq is 

 satisfactory for values of a smaller than approximately 

 half the value tanp/(2 -i- tan^P). In both cases, the upper 

 bounds for the function s^(o) depicted in figure 17 are 

 satisfactory for the range of small values of a for which 

 the steepness s_^(o') is large. It is noteworthy that these 

 upper bounds for the steepness of the divergent waves are 

 valid for all Froude numbers, since equations (12), (32) 

 and (33), and the upper bounds for |Kn^(t)| and |KQ(t)| do 

 not involve the Froude number. 



Figure 18 depicts the steepness function s^(o) of 

 the divergent waves corresponding to the slender-ship 

 approximation KQ(t) for the simple bow shape considered 

 previously in the two cases when the maximum flare angle 

 y is taken equal to and 45° (at the top and bottom 

 halves of the figure, respectively) and for two values of the 

 Froude number, namely for F = 0.8 and 0.3 (on the right 

 and left halves of the figure, respectively). Comparison of 

 the right and left halves of figure 18 shows no appreciable 

 difference between the values of the steepness function 

 s^(o) for the range of small values of a for which the 

 steepness is large, in agreement with the previously-noted 

 result that the upper bound for s^(o) depicted on the 

 right side of figure 17 is independent of the Froude 



iiat^o 



1/2 >^ 



Fig. 18 — Steepness of the Divergent Waves in the Kelvin 

 Wake for a Simple Ship Bow Form with /3 = 12°, y = 



(Top) and 45° (Bottom), and F = 0.3 (Left) and 0.8 

 (Right) in the Zeroth-Order Slender-Ship Approximation 



15 



