and (24b) show that the free-surface elevation within the 

 Kelvin wake at a sufficiently-large distance behind the ship 

 is given by 



(n/2)'^Vx)'''^x,Q) ~ RelAjexpKicp J + A + exp(i(p ^ )), (28) 

 where the phases <p ^ of the two waves are given by 

 (p^ =xe^ ±n/4. Curves along which the phases <p^ or 

 (p_ are constant then are defined by the equation 

 x6 ±n/4 = constant. This relation and equation (8) then 

 yield the following parametric equations for the curves 

 along which the phase is equal to - 2nn: 

 -x^ = (2n7t±n/4)/e^, (29a) 



y^ = (2nn±n/4)a/0^, (29b) 



where < o < 1/2^^^. The ten constant-phase curves 

 corresponding to 1 < n < 10 are depicted in figure 6. The 

 "transverse" and "divergent" waves in this classical 

 representation of the Kelvin wake correspond to the waves 

 A,"exp(i(p ) and A|^exp(iip^), respectively, in equation 

 (28). 



Fig. 6 — The Classical Kelvin Ship Wave Pattern 



The wavenumber corresponding to the wave with 

 phase cp^ is given by V<p^.. The corresponding 

 wavelength, say A^, and direction of propagation with 

 respect to the track of the ship, say /3^, then are given by 

 A^ = 2n/|V(p^.| and ft^ = tan" '((p*/<p^-), where cp* and 

 (p * represent the x- and y-derivatives of cp ^. and |V(p ^ | 

 = [(«Pjf )^ + (<Py*)^]'^^- The relation tp^ = xO^. ±n/4 and 

 equation (26c) then yield 

 A^ =2'^2,6„„2/(3^(,_8„2)l/2][,^2±(1.8^)l/2]l/2_ (30a) 



P^ = sin'{[l-»a^±(l-8«^)'^V[l+4<»^±(l-8<'^)'^^l}'^^- (30b) 

 Equations (30a,b) show that we have A _ = 2n, |3 _ = 

 and A^ = 0, /3^ = n/2 in the limit a = 0, and A_ = 

 4n/3 = A^, p_ = sin-'(l/3'^2) = p^ for a = X/l^''^. 

 More precisely, we have 



l-n>l_> 4n/3 > A^ > and (31a) 



0</3_ < sin-'(l/3'^^)</?+ <ii/2, (31b) 



as may be seen from figure 7 where the functions 

 A^(o)/2n and 2p_^(o)/n are depicted. 





■~._,i-«« 







— 2/3 



--^"^ 





— (2/.»*r'(l/t^) 



1 1 



X+/2ii/ ' 

 1 





0.1 



Fig. 7 — The Wavelengths Aj(o) and Propagation Angles 



/3 + (o) of the Transverse and Divergent Waves in the 



Kelvin Wake 



Equation (28) shows that the amplitudes of the 

 transverse and divergent waves in the Kelvin wake are 

 asymptotically given by (2/n)'^^|A,± |/(-x)'^^ as x — -<». 

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

 s^= (2/n)''2|Af |/(-x)'^^A^ ■ Equations (24a), (25a,b) 

 and (20b) then yield 

 (-x)^'h^ ~ o^(o)|K(t^)| as X - -«, (32) 



where o + (o) is defined as Q./n)^'\\+\^J''^/(+Q\i''^\^. 



Equations (26b,d) and (30a) then yield 

 O^ = [3 + (l-8o2)'^2][i_4„2^(,_g„2)l/2]l/2 



[l+4a2±(l-8o2)'/2]3/4/ 

 (AnO>'\')^'''cP'\\-icP-i^'\ (33) 



Equations (12) and (33) yield t_ = and o_ = 

 l/n(2Ti)'^^ in the limit o = 0, for which we have A_ = 2n 

 and p _ = as was noted previously. The steepness of the 

 transverse wave at a point (x,0) on the track of the ship 

 then is given by 

 s_(x,0) ~ |K(0)|/n(2n)'^2(-x)'^2 as x - -«. (34) 



Equations (12) and (33) also yield t^ ~ l/2o and 

 o_^~ l/16n'^^o''^^ in the limit o — 0, for which we have 

 A^— and /J^.-' n/2. We thus have o^~ x'l^/nQ.-n)''''- as 

 o -» 0, and equation (33) shows that the steepness of the 

 divergent wave at a point (x,o) in the vicinity of the track 

 of the ship is given by 

 s^(x,a) ~ t^^iKd j|/„(2n)l/2(-x)''2 



as o -* 0, with t^ ~ l/2o. (35) 



The steepness s^(x,o) then becomes unbounded as 

 cr - if 

 ("'^\VJiS')\ — oo as t - «. (36) 



