given by 2n^A,, vanishes if n^= 0, that is if the 

 waterline has a cusp at the bow or stern. Equation (S3) 

 shows that the first-order approximation to the function 

 Ag J also vanishes if the bow or stern is round, since we 

 then have Oy = and |, = -+„ = by symmetry. It 

 may thus be seen that the contribution of the bow or 

 stern to the far-field wave-amplitude function K(t) is of 

 order F^ if n^ = or n = at the bow or stern, 

 respectively; that is, we have 



Kg s = 0(F2) if n^ = or Uy = (56) 



at the bow or stern. In the particular case when the hull 

 surface is vertical at the bow or stern we have n^ = 

 and equation (53) becomes 



A, = t^-+,-ipt,V(l-pV) 'f "z = 0. (57) 



which yields 



A, = t,-+, + 0(F2) ifn^ = (58) 



as is indicated by the free-surface boundary condition |^ 



= -F'+xx = 0(F2). 



The contribution K^ of a point of the mean 

 waterline c where the phase of the exponential function 

 Eq* is stationary, that is where we have 



dyo/dxo = +l/t, (59) 



may be expressed in the form 



K^ = ±v(2nr)'''2A^ 



exp [+iv2(yot, - Xotj,)/ty2 + ien/4] , (60) 



where r is the radius of curvature of c at the point of 

 stationary phase (xQ,yQ), t is equal to + 1 or - 1 if the 

 center of curvature of c at the point (Xg.yg) is upstream 

 or downstream from (XQ.yg), respectively, and the 

 ampHtude-function A^ is given by 



A^ = (l-n^2)'^2(n^ty + .fd)-F^qA2=^ + 0(F''); (^1) 



in this expression, the second-order amplitude-function 

 A,* is defined by the equation 



±2[(xo')^ + (yo')^l'^^A2* = [(u^a,*)Ve^"]' 



+ u^a,*[5(e^"')V3e^"-9/>]/4(e^")2 



-i-2iq(u^)2a2*, (62) 



where the superscript ' denotes differentiation with 

 respect to the parameter A and the functions u^, a,*, aj* 

 and fl^ are given by equations (39), (40), (44) and (55), 

 respectively. 



The expression for the second-order amplitude- 

 function A2''^ is a complex one. However, equation (60) 

 and the first-order approximation to the amplitude 

 function, namely 



A± = (1 - n,^)'^^(n,ty + Id) + 0(F2), (63) 



provide a simple explicit expression for the stationary- 

 phase contribution K^ in terms of the geometric 

 characteristics of the hull and the downward tangential 

 derivative |j of the potential at the point of stationary 

 phase. In the particular case when the hull surface is 

 vertical at the point of stationary phase we have nj=0 

 and equation (63) becomes 



A^ = -+, + 0(f2) ifn^ = 0, (64) 



which yields 



A^ = 0(f2) if n^ = 0. (65) 



Equation (60) then shows that the stationary-phase 

 contribution at a point (xQ,yQ) of c where the hull is 

 vertical is of order F, that is we have 



K^ = 0(F) if n^ = 0. (66) 



On the other hand, equations (60) and (63) show that we 

 have 



K^ = 0(l/F) if n, * 0. 



(67) 



The stationary-phase contribution at a point where the 

 hull has flare thus is dominant in the zero-Froude- 

 number limit. 



The summation in equation (50) is extended to all 

 the points of the mean waterline c where the phase of 

 the exponential function Eq"*" or the function Eq" is 

 stationary, that is the points where the slope dyp/dx^ of 

 c is equal to - 1/t or + 1/t, respectively. The number of 

 stationary points, and their position along the waterline, 

 depend on the value of t and on the shape of c. For 

 instance, for the simple case of a hull with waterline 

 consisting of a sharp-ended parabolic bow region 1/4 < 

 X < 1/2 defined by the equation y = 4bx(l -2x), where 

 b denotes the ship's beam/length ratio, a straight parallel 

 midbody region - 1/4 < x < 1/4, and a round-ended 

 elliptic stern region - 1/2 < x < - 1/4 defined by the 

 equation y = b[-2x(l +2x)l'^^, there is one point of 

 stationary phase in the stern region given by x = 



- [I + 1/(1 +4b^t2)'^^]/4, so that we have - 1/2 < x < 



- 1/4 for < t < 0° with x -* - 1/2 as t -» and x -* 



- 1/4 as t -* oo, and one point of stationary phase in the 



