Equations (21a) and (6) show that the amplitude of the 

 divergent waves, given by A^|K(t^)|/(-i)'^^, vanishes 

 at the track of the ship if we have 



t'^^K(t) - as t - ». (22) 



Furthermore, it is shown in Barnell and Noblesse (1986) 

 that the asymptotic approximation (6) is uniformly valid 

 at the track of the ship if condition (22) is verified, h 

 may be seen from equations (21b) and (15) that the 

 steepness of the divergent waves, given by 

 S+|K(t^)|/(-4)'^^, is unbounded at the track of the 

 ship if we have 

 t^^2|K(t)| - oo as t - =0. (23) 



Both conditions (22) and (23) can be satisfied 

 simultaneously if we have 



|K(t)| ~ l/f as t - « with 3/2 < ^ < 7/2. (24) 



In summary, the asymptotic approximation (6) 

 expresses the far-field wave pattern of a ship at a point 

 (to), with I « - 1 and < a < 19°28' , as the sum of a 

 transverse wave and a divergent wave. The phase 

 iQ^{a)±n/4 of these two waves are defined explicitly in 

 terms of | and o; and their amplitudes 

 A^(a)K(t^)/(-|)'^^ are given by the product of the 

 functions A^(a)/(-|)'^^, which are also defined 

 explicitly in terms of | and a, and the far-field wave- 

 amplitude function K[t^((»)], which depends on the speed 

 (Froude number) and the shape of the ship. The far-field 

 wave-amplitude function is now considered. 



3. BASIC EXPRESSIONS FOR THE FAR-FIELD 

 WAVE-AMPLITUDE FUNCTION 



The far-field wave-amplitude function K(t) may be 



conveniently defined in terms of the nondimensional 



near-field coordinates (x,y,z) = (X,Y,Z)/L, where L is 



the length of the ship. In the Neumann-Kelvin theory, 



the function K(t) is given by the sum of an integral 



around the mean waterline of the ship and an integral 



over the mean wetted-hull surface. Specifically, for a 



ship with port and starboard symmetry, as is considered 



here, the function K(t) may be expressed in the form 



K(t) = K^(t) +'K_(t), (25) 



where the functions K^ are given by 



F^K^d) = r EJn/-Kt,+,-n^ty+j + iv2p+)ty dl 



+ v^ f. exp(v^p^z)E^(n^ + vWi"^) da, (26) 



as is shown in Noblesse (1983). In this equation, F is the 

 Froude number defined as 



F = U/(gL)'^^ 



V is its inverse, that is 



V = 1/F, 



p is defined as 



p = (l+tV'^ 



and E^ represents the exponential function 

 E = exp [-iv^p(x±ty). 



(27) 



(28) 



(29) 



(30) 



Furthermore, c and h represent the positive halves of the 

 mean waterline and of the mean wetted-hull surface, 

 respectively. The unit vector tangent to c and pointing 

 towards the bow is denoted by"f(tj|,t ,0), andTT(njj,n ,np 

 is the unit vector normal to h and pointing into the 

 water, as is indicated in Figure 2. The term n^ is defined 



Figure 2 - Definition Sketch for a Single-Hull Ship with 

 Port and Starboard Symmetry 



Also, dl and da represent the differential elements of arc 

 length of c and of area of h, respectively. Finally, 

 ^ = i^(x) represents the nondimensional disturbance 

 potential + = <t>/UL at the integration point Ton c or h, 

 1^, represents the derivative of + in the direction of the 

 tangent vector Tto c, and \^ is the derivative of + in the 

 direction of the vector TTxTi which is tangent to h and 

 pointing downwards as is shown in Figure 2. 



Equations (25) and (26) express the far-field wave- 

 amplitude function K(t) in terms of the value of the 

 Froude number and the form of the mean wetted-hull 

 surface, as was noted previously. More precisely, 

 equation (26) expresses the function Kj(t) as the sum of 

 a line integral around the mean waterline and a surface 

 integral over the mean wetted-hull surface of the ship. 

 Furthermore, these integrals involve the disturbance 

 velocity potential \ in their integrands. The relationship 

 between the far-field wave pattern of a ship and its form 



