the z axis pointing upwards, and the x axis is chosen in 

 the ship centerplane and pointing towards the bow. The 

 origin of the system of coordinates is placed within the 

 ship. The Froude number is denoted by F = U/(gL)'^^, 

 where L is the length of the ship. 



Equation (32) in [7] yields the following expression 

 for the velocity potential associated with the Kelvin wake 

 behind the ship 



n^(\) = Im 



/." 



[E^(t;x) + E_(t;x)]K(t)dt, 



(1) 



Im i 



where E^(t;x) is the exponential function 

 E^(t;T) = exp[z(l+t2) + i(x±yt)(l+t2)''2], (2) 



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

 depends on the hull shape and the Froude number. 

 Assuming that differentiation under the integral sign is 

 permitted in equation (1), we may obtain 



roo lE^ + E I I (l+t-)'''2 1 (3a) 



'O (e^-E_) |t(l-i-t-)'^M (3b) 



The nondimensional elevation e = Eg/U^ of the 

 free surface at a sufficiently-large distance behind the ship, 

 such that nonlinearities may be neglected, is given by 

 e(x,y) = a(t'(x,y,0)/ax. (4) 



The slopes of the free surface in the directions parallel 

 and perpendicular to the ship course then are 

 3e(x,y)/ax = d^(x,y,0)/ax- (4a) 



de(x,y)/dy = d-^(\,y,0)/d\dy. (4b) 



If differentiation under the integral sign in equation (3a) is 

 permitted, we have 



> I E _^ + E ^ I 1 1 + 1^ 



|e^-e_ 



The vertical velocity 4 is given by 



-'0 



K(t)) , dt. 



1 Itd+t^)) 



(5a) 

 (5b) 



In this study, we are mostly interested in the value 

 of the several fiow variables defined above at the mean 

 free surface and behind the ship, so that we have z = 

 and X < 0. Expression (2) for the exponential function E ^ 

 then becomes 



E^ = exp(ix(l + (jt)(l-i-t2)'^'l, (7) 



where a is defined as 



o = -y/x. (8) 



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

 considered here, the Kelvin wake is symmetric about the 

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

 Kelvin wake to the domain y > and x < 0, and assume 

 o > 0. 



Two difficulties associated with the foregoing 

 approach for numerically determining the potential, its 



gradient, and the free-surface slopes in the Kelvin wake 

 are readily apparent and should be noted here. A first 

 numerical difficulty stems from the oscillations of the 

 exponential function E^ given by equation (7), which are 

 very rapid for large values of |x|. We have x = Xg/U^ 

 = (X/L)/F^. For a typical value of the Froude number 

 equal to 0.2, say, we thus have x = 25 X/L; we then 

 have X = -250 at 10 ship lengths behind the ship, and 

 much larger values of |x| must be considered at greater 

 distances behind the ship or/and for smaller values of the 

 Froude number. Even for the comparatively-moderate 

 value of X equal to -50, figure 1 shows that the 

 functions E^ and E_ oscillate quite rapidly. More 

 precisely, figure 1 depicts the real parts of the functions 

 E^(t;x,<») and E_(t;x,o) for x = -50; o = .1, .2, 

 1/2^^2^ 4 anj 5. and for < t < 7 and < t < 3 on 

 the left and right sides, respectively. Figure I also 

 indicates that the behavior of the function E^(t;x,o) 

 strongly depends on the value of a. 



E+ (x = -50) E_ 



Fig. 1 — Real Parts of the Functions E + (t;x,a) and 



E_(t;x,o) for x = - 50 and o = .1, .2, 1/2^^^, .4 and .5 



A second, more basic, difficulty is associated with 

 the differentiation under the integral sign which was used 

 for obtaining expressions (3a, b) and (5a, b) from 

 expression (1). For a fully-submerged body, the far-field 

 wave-amplitude function K(t) is exponentially-small as t -* 

 00, so that the operation of differentiating under the 

 integral sign in expression (1) can be continued indefinitely 

 in principle. Differentiation under the integral sign likewise 

 is justified if z < 0. However, the operation must be 

 justified in the limiting case z = for a surface ship. 

 Clearly, the operation may not be justified in principle, or 

 feasible in practice, if the far-field wave-amplitude 



