7 











~^- 



""^""--..^lao 



» - ir, Y - 48* 







----,^5 



•to 



, 





^^^^ 



\\^^ 





^~^ 



1/7 



^^"^--^^^X 









1 



1/15 



""^--..^^^^ \s 





1/7 



^~~\^ 







M 



-3 



90 





X— .- 



Fig. 21 — Lines Along Which the Steepness of the 

 Divergent Waves is Equal to 1/20, 1/15 and 1/7 for a 

 Simple Ship Bow Form, with p = 12° and y = 45°, in 

 the Michell Thin-Ship Approximation ( ) and the 



Zeroth-Order Slender-Ship Approximation ( ) 



t 



n 



-— - 





::::\ 



^ 



IS 



t 

 n 









^^ 





-1 



900 



-800 -600 -400 



-200 







Fig. 22 — Lines Along Which the Steepness of the 

 Divergent Waves is Equal to 1/15 for a Simple Ship Bow 

 Form with p = 8°, 12°, 16° and y = 45° (Top) and 

 / = 0, 25°, 45° and /3 = 12° (Bottom) in the Zeroth- 

 Order Slender-Ship Approximation 



by the Michell thin-ship approximation K^(t) and the 

 zeroth-order slender-ship approximation KQ(t). This figure 

 strongly suggests the need for performing additional 

 calculations based on a more realistic mathematical model 

 than the simple thin-ship and slender-ship approximations 

 used in this study. 



Finally, the effect of the entrance angle p and of 

 the flare angle y on the steepness of the divergent waves is 

 illustrated in figure 22. Sf)ecirically, this figure depicts the 

 lines along which the steepness of the divergent waves, as 

 predicted by the slender-ship approximation Kfl(t), is equal 

 to 1/15 for p = 8°, 12°, 16° and y = 45°, in the top 

 half of the figure, and for y = 0, 25°, 45° and /3 = 12° 

 in the bottom half of the figure. This figure shows that 



the short divergent waves in the Kelvin wake become 

 steeper as the entrance angle p increases and as the flare 

 angle y decreases. More generally, figure 22 shows that the 

 short divergent waves in the Kelvin wake are strongly 

 influenced by the hull shape, and it therefore suggests the 

 need for performing additional calculations in whioh 

 systematic variations in hull shape are considered. 



SUMMARY OF RESULTS AND CONCLUSIONS 

 Asymptotic expressions for determining the velocity 

 potential and its derivatives at a sufficiently-large distance 

 behind a ship advancing at constant speed in calm water 

 are given by equations (23), (23a-e), (24a,b), (25a-d), (20a- 

 e) and (26a-d). The far-field asymptotic approximation 

 (23) is uniformly valid in the vicinity of the track of the 

 ship a = if condition (27) is satisfied. For the simple 

 bow shape considered in this study, condition (27) is 

 satisfied for k = and 1 , corresponding to the potential 

 ^ and the free-surface elevation ^^, when the far-field 

 wave-amplitude function K(t) is approximated by the 

 Michell thin-ship approximation K|^(t) or the zeroth-order 

 slender-ship approximation KQ(t). However, condition (27) 

 is not satisfied for k > 4 and k > 2 for the 

 approximations K,^ and Kq, respectively. 



The asymptotic approximations used in this study 

 provide simple explicit analytical expressions for 

 determining the velocity potential and its derivatives for 

 large values of - Xg/U^, that is in the far field, in terms 

 of the far-field wave-amplitude function. However, for 

 small and intermediate values of Xg/U^, these asymptotic 

 approximations are not useful, and the integrals (18) must 

 be evaluated numerically. For intermediate values of 

 Xg/U^, the exponential function E^(t;x)-HE_(tix) in the 

 integrands of the integrals (1), (3a, b) and (5a, b) oscillates 

 fairly rapidly, as may be seen from figure 1. Accurate and 

 efficient integration rules suited to oscillatory integrands of 

 the type depicted in figure 1 must be used. For small 

 values of Xg/U^, on the other hand, the oscillations of 

 the exponential function E^(tJx)-t-E (tjx) are not 

 significantly more rapid than the oscillations of the far- 

 field wave-amplitude function K(t) which also appears in 

 the integrands of the wave integrals (18), so that a 

 different integration rule is required. 



The amplitude, a^, of the divergent waves in the 

 Kelvin wake vanishes at the track of the ship if condition 

 (37) is satisfied. However, it is well known that the 



17 



