K^, respectively. Figure 14 depicts the amplitude functions 

 A^" and A^"* corresponding to + for the approximations 

 K„ and Kq. 



t " 



--°o^-y — \- 



-0.07, 



»-«•. 



■»- 



4r, F - 04 





k-0 





♦ 



k« 1 





— -A/^- /^~\r 





-d 



A / \ 





-- -V \ 



"• ' ♦ ' \\ 





-"- "^ 





m '4^ 





*«<♦»' 





*/ 





11 



'a. / 





■^--^''-y 





-^ 



r^-i 









\j ^---^ 





0.1 



0.2 



ta/i' 



villi 



0.1 0.2 0.3 ^a/i 0-^ ^-3 "-* 1/2r% 



Fig. 12a — Real (Top) and Imaginary (Bottom) Parts of 



the Functions K + a|f /( + e±)'^^ for k = (Left) and 1 



(Right), K = K|^ (Michell Thin-Ship Approximation) and 



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



F = 0.5 



Fig. 13a — Real (Top) and Imaginary (Bottom) Parts of 



the Functions K^.a|±/( +e;^)'^2 for k = (Left) and 1 



(Right), K = Kq (Zeroth-Order Slender-Ship 



Approximation) and a Simple Ship Bow Form with 



p = 12°, y = 45° and F = 0.5 



Fig. 12b — Real (Top) and Imaginary (Bottom) Parts of 



the Functions K^af /( + e:;)'''2 for k = 2 (Left) and 3 



(Right), K = Km (Michell Thin-Ship Approximation) and 



a Simple Ship Bow Form with p = 12°, / = 45° and 



F = 0.5 



It may be seen from figures 12a and b, 13a and b 

 and 14 that the amplitude functions A|^(o) and A|^(o) 

 become unbounded in the limit o -♦ 1/2^^^. This 

 singularity at o = 1/2^^^ stems from the fact that the 

 asymptotic approximation (23) is not uniformly valid in 

 the limit a — 1/2'^^, as was already noted. The amplitude 

 functions A^* (a) corresponding to the system of divergent 

 waves in the Kelvin wake also become unbounded in the 

 limit o -* in figures 13b and 14 corresponding to k = 2 



Fig. 13b — Real (Top) and Imaginary (Bottom) Parts of 



the Functions Kia^/(+e'^)"/2 for k = 2 (Left) and 3 



(Right), K = Kq (Zeroth-Order Slender-Ship 



Approximation) and a Simple Ship Bow Form with 



p = 12°, y = 45° and F = 0.5 



and 3 for the slender-ship approximation Kq and to k = 4 

 for both approximations K^ and Kq, respectively. This 

 singularity at o = illustrates the previously-noted 

 conclusion that the asymptotic approximation (23) is not 

 uniformly valid in the vicinity of the track of the ship a 

 = if condition (27) is not satisfied. Equations (20a-e) 

 show that we have a^ ~ 1, a, ~ t, aj ~ t^, aj ~ t^ and 

 a. ~ t' as t -* <», and equations (45) and (67b) yield 

 |K|^| ~ 1/t' and IK^I ~ 1/t^ as t -* <», respectively. 

 Condition (27) therefore is not satisfied for k > 4 and k 

 > 2 for the approximations K^ and Kq, respectively. 

 Condition (27) however is satisfied for k = and 1, 



13 



