corresponding to the potential j and the free-surface 

 elevation j^, for both the approximations K^^ and Kq. 



Figures 12a and 1 3a show that the amplitude 

 functions A^ and Aj", corresponding to the transverse 

 waves in the Kelvin wake, generally are larger in 

 magnitude than the amplitude functions A^ and Aj*' 

 corresponding to the divergent waves, whereas the reverse 

 may generally be seen to hold in flgures 12b and 13b for 

 the amplitude functions A^ and A3* . The relative 

 importance of the divergent waves with respect to the 

 transverse waves thus increases with k. Indeed, the 

 transverse-wave amplitude function A^ia) is hardly visible 

 on the scale of the divergent-wave-amplitude function 

 A^{a) used in figure 14. 



0.3 ,^>5 



1/21^ 



Fig. 14 — Real (Top) and Imaginary (Bottom) Parts of 



the Functions K^af/{ +e±)'^2 for K^, (Left) and Kq 



(Right) and a Simple Ship Bow Form with p = 12°, 



Y = 45° and F = 0.5 



The divergent-wave-amplitude functions A|^(o) 

 associated with the approximation Kq are most notably 

 different from the corresponding functions A|^(o) 

 associated with the Michell approximation Kj^ for values 

 of a in the vicinity of o = 0, as was already noted, and 

 of a = tanp/(2 -I- tan^/3). In the vicinity of this value of o, 

 the divergent -wave-ampUtude functions A|^(o) associated 

 with the approximation Kq exhibit a peak, which is quite 

 pronounced for k > 1. The foregoing particular value of 

 a corresponds to the special case when the point of 

 stationary phase t^, defined by equation (12), is equal to 

 the value 1//3q = 1 /tan/3 for which the function KQ(t) 

 displays a peak, as may be seen from figures 10 and 11. 



Figure 15 depicts the amplitude functions 

 (2/n)'^2A,- and (2/n)'^2A,+ that are associated with the 

 free-surface elevation far behind the ship, as is specifically 



P - ir, Y - 4C. F - o.s 



2 + tin*9 



\a<n. 



Fig. 15 — Amplitude of the Transverse (-) and 



Divergent (-t-) Waves in the Kelvin Wake for a Simple 



Ship Bow Form, with /3 = 12°, y = 45° and F = 0.5, in 



the Michell Thin-Ship Approximation (Left) and the 



Zeroth-Order Slender-Ship Approximation (Right) 



indicated in equation (28), for the previously-considered 

 ship bow shape. It may be seen that the amplitude a^(a) 

 of the divergent waves in the Kelvin wake vanishes as o -• 

 0, that is at the track of the ship, and is generally smaller 

 than the amplitude a_(o) of the transverse waves; this is 

 especially true in the vicinity of the track of the ship. 

 Differences between the wave-amplitude functions a^.(af) 

 corresponding to the approximations K|^(t) and KQ(t) are 

 particularly striking for the amplitude a_^(o) of the 

 divergent waves in the vicinity of a = tan/3/(2-l-tan /3), 

 where the function a^(a) associated with the 

 approximation KQ(t) exhibits a sharp peak. 



Figure 16 depicts the wave-steepness functions s_(a) 

 and s ^ (a), which correspond to the ratios of the wave- 

 amplitude functions a_(a) and a^(o) depicted in figure 15 

 over the wavelength functions A_(a) and A^(o) defined by 

 equation (30a); the steepness functions s^(a) are 

 specifically defined by equations (32) and (33). The 

 divergent waves in the Kelvin wake may be seen to be 

 generally steeper than the transverse waves, even though 

 figure 15 shows the transverse waves to be larger in 

 amplitude than the divergent waves. This is especially true 

 near the track of the ship where the steepness of the 

 divergent waves becomes infinitely large, even though 

 figure 15 shows that their amplitude vanishes as a -* 0. 

 The divergent waves become infinitely steep at the track 

 of the ship because the wavelength A^(a)-»0aso-*0 

 and condition (36) is satisfied, for both the 

 approximations K^(t) and Kg(t) as may be seen from 



14 



