2. STATIONARY-PHASE APPROXIMATION TO 

 THE FAR-FIELD KELVIN WAKE 



The far-field Kelvin wake may be conveniently 

 analyzed in terms of the nondimensional far-fleld 

 coordinates (tfj,{) = (X.Y.Z) g/U^, where g is the 

 gravitational acceleration, U is the speed of advance of 

 the ship, and (X,Y,Z) are dimensional coordinates. The 

 mean free surface is taken as the plane i = 0, with the 

 i axis pointing upwards, and the 4 axis is chosen along 

 the track of the ship, that is in the ship centerplane, and 

 pointing towards the bow. The origin of the system of 

 coordinates is placed within the ship. 



Equations (4), (3a), (7) and (8) in Barnell and 

 Noblesse (1986) yield the following expression for the 

 elevation i of the free surface at a sufficiently large 

 distance behind the ship, such that nonhnearities may be 

 neglected: 



(1) 



n«|,r,) = Re^°° (E^ +E_) K(t) (l-l-t^)'^^ dt, 



where K(t) is the far-field wave-amplitude function and 

 E^ is the exponential function defined as 



E^(t;4,o) = exp [i|e^(t;o)], (2) 



with the phase-function 9^ and the parameter o defined 

 as 



B^(t;o) = (I+ot)(l-i-t2)'^2_ (3) 



o = r)/(-l)- (4) 



For a ship with port- and starboard-symmetry, as is 

 considered here, the Kelvin wake is symmetric about the 

 ship track r) = 0. We may then restrict the analysis of 

 the Kelvin wake to the quadrant jj > 0, 4 < and 

 assume o > 0. Let a be the angle between the track of 

 the ship and the line joining the origin of the system of 

 coordinates to the observation point (4,r)). We thus have 



a = tan^'o. (5) 



A far-field asymptotic approximation, valid as | -* 

 -<», to the wave integral (1) may be obtained by 

 applying the method of stationary phase, as is well 

 known. The result of this classical asymptotic analysis 

 may be found in Barnell and Noblesse (1986), for 

 instance. Specifically, equations (28), (24a), (20b), (25a-d) 

 and (26a-d) in this reference yield 



(-4)l^2^(^o)'^^ Re 



{A_K(t_)exp[i(4e_-n/4)] 



-t- A^K(t^)exp [i(4e^-K7i/4)l} asi- -», (6) 



with Q K s <. tan ~''2~^'^^) o^: !9°28' 



where t^(a) are the values of t for which the phase- 

 function e+(t;o) is stationary, 6^(0) = e^(t ;<») 

 represents the corresponding values of the phase-function 

 8^(t;o), and A^(o) is the function defined as A^(o) = 

 (2/n)1^2(l-Ht^2)l''VlTe:;(t^;o)]"'2. The functions t^(a), 

 QJ.a) and A^(o) are given by 



t>) = [l±(l-8o2)'^2)/4„_ (7) 



e^(a) = [3T(1- 802)1/2] 



[l-l-4o2±(l-8o2)'^2]'^V2'''28o, (8) 



A^(<») = [l-(-4o2±(l-8o2)i/2j3/42l/4/ 



4n'/V^2(i_g„2)I/4_ 



where we have 

 o = tano 



(9) 



(10) 



as is given by equation (5). 



The far-field asymptotic approximation (6) shows 

 that the wave pattern at any point (|,a), with 4 <K - 1 

 and < a < 19°28', consists in two elementary plane 

 progressive waves, so-called transverse and divergent 

 waves, as is most well known. The wavelengths K^ and 

 the directions of propagation /3^, measured from the 

 track of the ship, of the transverse (A_,/3_) and 

 divergent (A+,/3^.) waves at an angle a from the track of 

 the ship are given by 



X^(a) = 2"2l6no2/[3+(l-8o'^)"'^] 



[l-4o2±(l-8o2)'/2]l/2_ (,,) 



PJa) = cos-'{2'^22o/[I+4o2±(l-8o2)'^2l'^2}. (12) 



We have 



1 » /l_/2n >2/3 > A^/2n >Oand (13) 



0< li_ < sin- '(l/3'''2) C^ 35°16' </?+=« 90°. (14) 



Equation (6) shows that the amplitudes of the 

 transverse and divergent waves in the Kelvin wake are 

 asymptotically equal to A^|K(t^)|/(-4)'^2 as 4 "* -"• 

 The steepnesses, say s^, of these waves then are given by 

 Sj = AJK{tJ\/kJ-i)^^^. We then have 



( - i)^'\a.<') - S^|K(t^)| as i - -00, (15) 



where the functions S^(o) are defined as S^(o) = 

 A^(o)/A^(o). Equations (9) and (11) then yield 



SJo) = [3T(l-8o2)'''2] [l-4o2±(l-8o2)'^2l'^2 



[1 +4o^±{l -So^)^'^]^"'/2^^*64n^'^o^^H\ -%oW>. (16) 



