and speed thus is a complex one, which offers little 

 physical insight. However, analytical approximations for 

 the waterline and the hull integrals in equation (26) can 

 be obtained in the limiting case when the value of the 

 Froude number is sufficiently small. This low-Froude- 

 number asymptotic approximation for the function K(t) 

 is now obtained. 



4. LOW-FROUDE-NUMBER LAPLACE 



APPROXIMATION TO THE FAR-FIELD WAVE- 

 AMPLITUDE FUNCTION IN TERMS OF A 

 WATERLINE INTEGRAL 



For large values of v = 1/F, or more generally of 

 vp, the exponential function exp(i^p^z) in the hull- 

 surface integral in equation (26) vanishes rapidly for 

 negative values of z. Therefore, only the upper part of 

 the mean-wetted hull surface h yields a significant 

 contribution in the low-Froude-number limit. More 

 precisely, the hull-surface integral can be approximated 

 by a line integral around the mean waterline c. 

 Furthermore, this integral can be combined with the 

 waterline integral in equation (26). The analysis is 

 presented in detail in Noblesse (1986a) and is briefly 

 summarized below. 



The rnean waterline is represented by the 

 parametric equations 



x = Xo(A) and y = y^W, (32a,b) 



where the parameter X varies between its bow and stern 

 values, that is 



X„ < A < Ao 



(32c) 



In the vicinity of the mean free surface, the hull surface 

 is represented by the parametric equations 



X = Xq(A)-i-z x,(X)-t-z^X2(A)-(-. . . , (33a) 



y = VoW + z y,(A)-t-z2y2(l) + (33b) 



where Ag « A < Ag and z < 0. (33c) 



The velocity potential +(A,z) on the hull surface in the 

 vicinity of the plane z = likewise is expressed in the 

 form 



^ = if^W + z^^(k) + z\W + (34) 



Differentiation of the functions Xj,(A), yn(A), i^„(A) with 

 respect to the parameter A is denoted by the 

 superscript ' ; thus, we have Xq = dxQ(A)/dA. 



By using the foregoing parametric representations 

 for the hull surface and the velocity potential, applying 



(35) 



(36) 



(37) 



(38) 



(39) 



the Laplace method for approximating the hull-surface 

 integral in equation (26), and combining the resulting 

 waterline integral with that already present in equation 



(26), we may obtain after lengthy algebraic 



transformations the following low-Froude-number 



approximation for the function K(t): 



K(t) -v v2q2 r^S(Eg + a^ +E(,-a_) dA as F - 0, 

 ^Ag 



where we have 



q = 1/p = 1/(1 -(-t^)'''^, 



Eq* is the exponential function 



Eq* = exp[-iv^p(Xo±tyo)), 



and a^ is the amplitude function 



a± = u^a,*-i-FV(ujV + 0(F^)' 



where u^ is defined as 



u^ = l/[l-iq(x,±ty,)], 



and the functions a,* and aj* are now defined. 



The first-order amplitude function a,* is given by 

 ^1* = yo'A^/(l+£2) + 2q(Xo'±ty(,')(u^)2B^+o 



-I- C^+o' + uDJ^/{\+t^) + ip(y|+o'-yo'+i). (40) 

 where we have 



£ = (yo'''i-Xo'y|)/u, (41) 



with u defined as 



u = [(Xo')2 + (yo')^l'^^ (42) 



and the coefficients A^, B^., C^ and D^ are defined as 

 A^ = I(H-pyoVu)(l-py(,'/u) + £2) 



+ i(pyo'/u)[y,(Xo'±tyo')/u + £], (43a) 



B± = <i(yr^^^2> + '(y,X2-x,y2), (43b) 



C^ = [(«-p%'yo'/u2) 



+ (pyo'/u)^«(XiXo' -i-y|yo')/u(l +£2)] 

 + i[y,(Xo'±tyo')/u + £) 



[pXfl ' /u - (pyQ ' /u)£(x , Xq ' + y , yg ' )/u( 1 + £^)] , (43c) 



D^ = ((X(,'±tyo')/uJ[(l-F£2)(y,±iqt)Uj-hi(pypVu)£y,l 



- (pyo'/u)£(pyoVu-i£). (43d) 



The second-order amplitude function aj* is given 

 by 



+ 6i(u^)2[+(,(m , ±r2* + mo-ys*) + + 1 V>'2*1 



- l2{u^)\moHY2^)\ (44) 



