where we have 







Xn* = '•("n^tyn)- 





(45) 



m„* = (in + iq(y„'Ttx„'). 





(46) 



with 







Mo = «". 





(47a) 



Ml = "iVr-yixr + 2(x2yo' 



-y2N)'). 



(47b) 



M2 = ''iy2'~yi''2' + ^(xjy,' 



-yz",') 





+ SCXjYo'-yjXo'). 





(47c) 



In the particular case when the phase of the 

 exponential function Eg^ is stationary, that is at a point 

 (xQ.yp.O) where we have Xo'±tyQ' = 0, the first-order 

 amplitude function a,- takes the form 



a,- = ± eliyg' +(py,+it)'^o'+u4.,]/(l±i£) 



+ ip(yi+o'-yo'+l) if Xo'±tyo' = 0. (48) 



Furthermore, if the hull surface intersects the plane z = 

 orthogonally at a point of stationary phase we have 



ar = iF2pyo'3\ax2 



if XQ'±tyo' = and n^ = 0, (49) 



and the amplitude function a^ then is or order F^. 



The low-Froude-number asymptotic approximation 

 to the far-field wave-amplitude function K(t) given by the 

 waterline integral (35) is considerably simpler than the 

 exact expression (26), which involves both a waterline 

 integral and a hull-surface integral; and the approximate 

 expression (35) is well suited for efficient numerical 

 evaluation. However, expression (35) can be simplified by 

 applying the method of stationary phase, which takes 

 advantage of the rapid oscillations of the exponential 

 function Eq- given by equation (37) in the low-Froude- 

 number limit V -► °o, or more generally in the limit 

 vp -* °°. This stationary-phase approximation is now 

 obtained. 



5. LOW-FROUDE-NUMBER STATIONARY-PHASE 

 APPROXIMATION TO THE FAR-FIELD WAVE- 

 AMPLITUDE FUNCTION 



The method of stationary phase indicates that the 



major contributions to the integral (35) in the limit when 



the exponential functions Eq- are rapidly oscillating, that 



is if V -* 0° or more generally i/p -* ■», stem from points 



where the phases of these exponential functions are 



stationary, that is from points where we have XQ'±tyQ' = 



0, and from the end points Ag and k^ of the integration 



range, that is from the bow and the stern. This 

 stationary-phase analysis is presented in detail in 

 Noblesse (1986b) and only its results are given here. We 

 have 



K(t) ~ iq^Kg - Kg -(- Z K^) as F - 0, (50) 



where Kg and Kg correspond to the contributions of the 

 bow and the stern, respectively, and K^ corresponds to 

 the contribution of a point where the phase of the 

 exponential function Eq^ is stationary, that is where we 

 have yo'/(-''Q') = ±l/t or dyg/dx^ = Tl/t; the 

 summation in the asymptotic approximation (50) is 

 extended over all such points of stationary phase on the 

 mean waterline c. The expressions for the contributions 

 of the bow and stern Kg ^ and of the points of 

 stationary phase K_^ are given below. 



The contribution Kg 5 of the bow or stern may be 

 expressed in the form 



■^B.S = Ag5exp(-ivVB,s)' (5') 



where Xg g is the abscissa of the bow or stern and the 

 amplitude-function Ag 5 is given by 



Ag s = 2n,A, + iF2q(A2 + -t- A^ " ) -1- O(F^); (52) 



in this expression, the functions A| and A2- are defined 

 by the equations 



(nj, + iqn^)A, = n^n^l 1 - ipn^(n^ -1- iqn^)/(l - n^^ - p\'^)] 



+ ty.^, - ipnyty+j(l - n^2 + ipn^n^)/(l - n^^ _ p2n^2) (53) 



and 



e^'Aj* = (u^a,Ve^')'-i-iq(uJ^*, (54) 



where the superscript ' denotes differentiation with 

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

 are defined by equations (39), (40) and (44), respectively, 

 and the function 6^ is defined as 



e+ = X(,±tyo. (55) 



Expression (54) for the second-order amplitude- 

 function Aj* is a complex one. However, equation (53) 

 defines the first-order amplitude function A, explicitly in 

 terms of the value of t, p = (1 +t^)'^^ and q = 1/p, the 

 geometrical characteristics of the hull at the bow or 

 stern, namely the unit vectorT(t^,ty,0) tangent to the 

 mean waterline and the unit vector n'(njj,ny,nj) normal 

 to the hull, and the components ^, and ^^ of the velocity 

 vector in the directions of the unit vectors Fand n"xT 

 tangent to the hull. Equation (52) shows that the first- 

 order approximation to the amplitude-function Ag 5, 



