integrals deflned as 



Vk* ("•") = f^ «Pl*''« r C^-' )lK(tKW dt: (18) 



in this expression, tlie ptuue B^(t;a) is denned by 

 O^iUa) = (iTotXl +t^)'^^ with o > 0; (19) 



finally, the constants c^ in equation (17) and the 

 functions A^^(t) in equation (18) are defined in the 

 following table: 



k 4^ a,^ c^ 



i 1 1 



1 ♦, (l+t2)'^2 i 



2 i t(l+t2)'^2 i 



(20a) 



(20b) 



i (20c) 



3 +„ l+t^ -1 -1 (20d) 



4 l^y t(l+t2) -11 (20e) 



In the particular case o = 0, that is on the track of 

 the ship, equation (18) yields i^j^ = h>^^= v^ , with 

 Vk(x,0) = yj,°° exp[ix(l+t2)"'2] K(t)a^(t) dt. 

 The major contribution to this integral in the limit x -» 

 -op stems from the point of stationary phase at the origin 

 t = 0. Specifically, we may obtain 

 (n/2)'^2(_x)i/24l^(x,0)~ 



Im C|jKq exp[i(x - n/4)] as x -» -°°, (21) 



with Cq = 1, c, = i, Cj = - 1, (21a,b,c) 



and C2 = = c^; we thus have 



+y = = 4^y for a = 0, (22a,b) 



in accordance with the symmetry of the wave pattern 

 about the axis o = 0. In equation (21), K^ represents the 

 value of the function K(t) at the origin t = 0, that is we 

 have K<, = K(0). 



For < o < 1/2'^^, the phase B_^,(t;a) is stationary, 

 that is e;^ = 0, at the two distinct points t_(a) and t^(a) 

 defined by equation (12), whereas B'_{V,a) > for t > 0, 

 as may be seen from figure 3b. Equations (17) and (18) 

 then yield 

 n+^(x,<») ~ Im c^v^^^ as X - -00 with 0< a < \ /!?'''■ . 



The contribution of the two points of stationary phase t ^ 



can be evaluated by using the method of stationary phase, 



with the result 



(n/2)'^2(-x)'''\(x,a) 'v Im c^(A^^E_ + A + E^) 



as X — -00, with < o < 1/2^^^ and (23) 



Cq = 1, C| = i = Cj, Cj = - 1 = c^; (23a,b,c,d,e) 



furthermore, A^ and E^ are the amplitude and 



exponential functions defined as 



A± = K^a±/(+e;)'^2_ (24a) 



E^ = exp[i(xe^±Tr/4)], (24b) 



where K^, a^, 9^ and 6'^ are defined as 



K^= K(t^), a±= a^(t^), (25a,b) 



e ^ = 8(t ^ ;a), e; = e'(t ^ \a\ (25c,d) 



and t^ is the function of a given by equation (12), that is 



we have 



t^ = [l±(l-8<»2)'^^]/4<.. (26a) 



We may then obtain 



l+t±^ - [l+4o^±(l-8o2)'^V8o^- (26b) 



The expressions for the terms a^ may readily be obtained 



from equations (20a-e), (25b) and (26a,b). Equations (2Sc), 



(19) and (26a,b) yield 



e^ = (3T(l-8o2)'^^l[l+4<»2±(l-8o^)'/2]l/2/2'/2g„ (26c) 



Finally, 6^ is given by equation (15a), that is we have 

 Te; = 2'^Vl-8o2)'''V[l+4a2±(l-8o2)'^2]'^2. (26d) 



It may be shown from equations (26a,c,d) and 

 verified from figures 4, 3a, 3c and 5 that we have t _ = 

 and 9_ = 1 = ei_ in the limit o = 0. Furthermore, 

 equations (25b) and (20a-e) show that we have a^ = 1 foi- 

 k = 0, 1 and 3, and a^ = for k = 2 and 4 in the limit 

 = 0. The asymptotic approximation (23) for \^x,a) 

 therefore becomes identical to the asymptotic 

 approximation given by equations (21), (21a,b,c) and 

 (22a,b) for +|((x,0) in the limit a = 0, if the contribution 

 of the second point of stationary phase t ^ = <» is null, 

 that is if A|^ = for o = and t^ = <». In other words, 

 the asymptotic approximation (23) for l^^,a) is uniformly 

 valid in the vicinity of the track of the ship o = if A,^ = 

 for o = 0. Equations (26a) and (26d) yield t^'^' l/2a 

 and -0'^ ~ 2o as o -* 0. We then have -0'^ ~ 1/t^ 

 as o -* 0, and the condition for the asymptotic 

 approximation for ^j,(x,o) to be uniformly valid in the 

 limit a = takes the form 

 t'''2K(t)a^(t) - as t ^ =«. (27) 



In the limit a = 1/2^^^, we have t_ = 1/2'^^ =- 

 t and 0'^ = 0, as may be verified from equations 

 (26a,d) and figure 5. Equation (24a) then shows that 

 have |A|r| -» °° as a -* 1/2^''^; and the asymptotic 

 approximation (23) is not valid in the vicinity of 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 [11] for the particular case of a pressure point at 

 the free surface. However, we are mostly interested in the 

 sector < a < 1/2'^^, that is inside the Kelvin wake, in 

 the present study. 



The far-field asymptotic approximation (23) shows 

 that the wave pattern at any point (x,a), with x <S; - 1 

 and < a < 1/2^^^, consists in two elementary plane 

 progressive waves. Specifically, equations (4), (20b), (23) 



