Equations (7), (8), (9). (10) and (16) show that we 

 have 



t_= 0, e_= 1. A_= (2/ii)'^2 

 and S_ = 1/2'^V^^ for a = 0. 

 t^~ l/2a, e^~ \/4a, A^~ l/2n'''V^^ 

 and S^~ l/16n^^2<»^^2 as • - 0, 

 t^ = 1/2'^^ and 9^ = 3(3/2)'''V4 

 for o = tan" '(1/2'''^) ~ 19°28', 

 A^ ~ S/n'^^e'^^d -8o2)'/< and 

 S^~9/4n3^Z6'^'»(l-8a^)'^'» 

 aso- tan-'(l/2'^2). 



(17a,b,c) 

 07d) 



(18a,b,c) 



(18d) 



(19a,b) 



(19c) 

 (IM) 



The stationary-phase values t^(a), the phase-functions 

 ©±(<»), the amplitude-functions A^(o) and the steepness- 

 functions S^(a) are depicted in Figures la and lb, which 

 correspond to the transverse and divergent waves, 

 respectively. 



It may be seen from these figures and from 

 equations (19c) and (19d) that the amphtude-functions 

 A^(o) and the steepness-functions S^(o) are singular at 

 the Kelvin cusp line o^ = 1/8, a = tan" '(1/2^^^) ~ 

 19°28', in accordance with the well-known fact that the 

 asymptotic approximation (6) is not uniformly valid at 

 the boundary of the Kelvin wake. A complementary 

 asymptotic approximation, expressed in terms of Airy 

 functions, valid at and near the Kelvin cusp line is given 

 in Ursell (1960) for the particular case of the Kelvin wave 

 pattern due to a concentrated pressure point at the free 

 surface. However, Ursell's more complex asymptotic 

 approximation will not be considered here because the 

 simple asymptotic approximation (6) is little affected by 

 the weak singularity (1 -8o^)" '''' for points (4,o) inside 

 the Kelvin wake and not too near the cusp line a :u 

 19°28', in which we are mostly interested in this study. 



Equation (11) yields 



A ^ ~ ina^ as o — 0. (20) 



This approximation and the approximation (18b) show 

 that there are an inflnite number of divergent waves with 

 indeflnitely shorter wavelength in the vicinity of the track 

 of the ship, as is well known. It may be seen from 

 Figure lb and equations (18a,c,d) that the stationary- 

 phase value t^(o), the amplitude-function A^(o) and the 

 steepness-function S^(a) are unbounded in the limit a -* 

 0. Equations (18a,c,d) yield 



A - 



1.6- 







i 



1.2- 



e_ 



^ 



/ 



J 









u.o- 



A_ 



0.4- 





- T r 



i 



0- 



^"""Y^ -r 1— 



6» 9' 



a 



Fig. la - The Stationary-Phase Value t_(o), the Phase- 

 Function 9_(o), the Amplitude-Function A_(o) and 

 the Steepness-Function S _ (cr) Corresponding to the 

 Transverse Waves in the Kelvin Wake 



a 



Fig. lb - The Stationary-Phase Value t^(<»), the Phase- 

 Function e^(o), the Amplitude-Function A^(a) and 

 the Steepness-Function S^(<») Corresponding to the 

 Divergent Waves in the Kelvin Wake 



A+ ~ (2/n)'^^^'^^ and 



S+ ~ t^^^V2'^V^^ as t^ - ». 



(2U) 

 (21b) 



