Fig. 19 — Amplitude of the Peak Value of the Steepness 



of the Divergent Waves in the Kelvin Wake for a Simple 



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



Zeroth-Order Slender-Ship Approximation 



number. Comparison of the top and bottom halves of 



figure 18, corresponding to y = and 45° as was already 



noted, shows appreciable differences, especially for F = 



0.3 for which the function s_^(o) exhibits a very 



pronounced peak in the case y = 45°. No such peak is 



apparent in the top half of the figure for y = 0, that is 



in the case when the hull intersects the free surface 



orthogonally. 



The right half of figure 16 and the bottom half of 



figure 18 show that the magnitude of the peak in the 



steepness function s^(c») for a = tan/3/(2 + tan^|3) strongly 



depends on the value of the Froude number. Specifically, 



the peak is very pronounced in figure 18 for F = 0.3, 



fairly pronounced in figure 16 for F = 0.5, and almost 



nonapparent in figure 18 for F = 0.8. The magnitude of 



the peak in the steepness function, that is the value of the 



function s^(o) for a = tan/3/(2 + tan^P), is represented in 



figure 19 as a function of the Froude number, which is 



based on the length of the bow region. This figure shows 



that the magnitude of the peak increases very rapidly as 

 the Froude number decreases below a certain threshold 

 value in the vicinity of F = 0.3. 



Figure 20 depicts the boundary of the Kelvin wake, 

 which corresponds to o = -y/x = 1/2'^^ (that is, an 

 angle equal to approximately 19°28'), the line a = 

 tan/?/(2 + tan^/?) ~ p/2 (that is, an angle equal to 

 approximately 6°) along which the steepness of the 

 divergent waves has a peak, and the lines along which the 

 steepness of the divergent waves is equal to 1/20, 1/15, 

 and 1/7 (shown as a chain line close to the track of the 

 ship). The latter three lines were determined by using the 

 upper bound for the steepness function s _^ (a) that was 

 determined previously and depicted on the right half of 

 figure 17. The four lines inside the Kelvin wake shown in 

 figure 20 correspond to the zeroth-order slender-ship 

 approximation ^(\) for the simple bow shape considered 

 previously, with /3 = 12°, y = 45° and d = 0.1. The 

 three lines along which the divergent waves are steep lie 

 much closer to the track of the ship than the lines a ~ 6° 

 along which the steepness of the divergent waves exhibits 

 a peak. 





$ = ly, -y » 45* 



KO 





""^---...^KELWN 





PEAK 



-^-rrr^ 



\^ 



00 









Fig. 20 — The Kelvin Cusp Line, the Line Along Which 

 the Amplitude of the Divergent Waves Exhibits a Peak, 

 and the Three Lines Along Which the Steepness of the 

 Divergent Waves is Equal to 1/20, 1/15 and 1/7 (Chain 

 Line Close to the Track of the Ship) for a Simple Ship 

 Bow Form, with fi = 12° and y = 45°, in the Zeroth- 

 Order Slender-Ship Approximation 



The three lines along which the steepness of the 

 divergent waves is equal to 1/20, 1/15 and 1/7, which are 

 depicted in figure 20 for > x ^ - 100 and < y < 40, 

 are represented again in figure 21 at a distorted scale 

 where > x > - 300 and < y < 7. The corresponding 

 constant-steepness lines predicted by the Michell thin-ship 

 approximation are also shown in figure 21 for 

 comparison. The latter lines were determined from 

 equations (32) and (33) and the upper bound |K,^(t)| < 

 ^PqA^ given by equation (45). Figure 21 shows significant 

 differences between, the constant-steepness lines predicted 



16 



