9 - 12*, F - O.B 



0.03 



Fig. 10 — Real and Imaginary Parts of the Functions 



K|^(t) and K^Ct) for a Simple Ship Bow Form with 



r = and 45°, p = 12° and F = 0.5 



slender-ship approximation K^(t) for values of t in the 

 neighborhood of 1/tan^. The influence of y upon |Kg(t)| 

 for large values of t is explicitly indicated by the 

 asymptotic approximations (67a,b). 



Finally, figure 1 1 depicts the real and imaginary 

 parts of the functions (1 +t^)'^^K^(t) and (1 +t2)'^2Ko(t) 

 for the simple bow shape depicted in figure 8, for which 

 we have d = 0.1, /? = 1-2° and / = 45°, at a value of 

 the Ffoude number F equal to 0.5. Differences between 

 the approximations K,^ and Kq may be seen to be quite 

 substantial. In particular, the function (1 +t^)'''^KQ(t) has 

 a peak at t = 1 /tan/3 = IZ/Sq. 



Fig. 11 — Real (Tap) and Imaginary (Bottom) Parts of 



the Functions KM(t) and Ko(t) for a Simple Ship Bow 



Form with p = 12°, y = 45° and F = 0.5 



A SIMPLE TEST CASE: 

 THE FAR-FIELD KELVIN WAKE 



The expressions for the far-field wave-amplitude 

 function K(t) obtained in the foregoing section for a 

 simple bow shape and for two simple approximations to 

 the function K(t), namely the Michell thin-ship 

 approximation K|^ and the zeroth-order slender-ship 

 approximation Kq, may now be used into the previously 

 determined asymptotic approximations for the far-field 

 Kelvin wake. 



Far behind the ship, that is for x -* -0°, the 

 functions iji|,(x,a), where < k < 4, defined by equations 

 (17), (18), (19) and (20a-e) are given by the asymptotic 

 approximation (23). The real and imaginary parts of the 

 amplitude functions A|['(a) and A|^(a) in this asymptotic 

 approximation, given by equation (24a), are depicted in 

 figures 12a and b, 13a and b and 14 for < a < 1/2'^^. 

 More precisely, figures I2a,b and 13a, b represent the 

 amplitude functions A^^ and \^ for k = 0,1,2,3 

 associated with the potential ^ and its derivatives 4,^, + 

 and ifijyj, and correspond to the approximations K^ and 



12 



