Pq = lanj), Yq = tany (42a,b) 



will be used for shortness hereafter. 



The far-field wave-amplitude function K(t) for the 

 foregoing semi-infinite hull has been evaluated for two 

 simple approximations defined explicitly in terms of the 

 hull shape and the Froude number, namely the Michell 

 thin-ship approximation and the zeroth-order slender-ship 

 approximation [7], for which the function K(t) is denoted 

 K|^(t) and Kg(t), respectively. The Michell thin-ship 

 approximation is given by the product of two single 

 integrals, as follows: 







/; 



explv^d-t-t-'jz] (/3o + y(,2)dz. 



These integrals can be evaluated analytically, with the 



result 



K^a) = 4[/3(,-(y3o-yod)e-F2yo(l-e)/(l+t2)] 



sin[v^(l-t-t2)'^V2] exp[-iv^(l-i-t^)'^V2]/(l-i-t^)^^^ (43) 

 where the term e is defined as 



e = expl-v^dd-i-t^)]. (44) 



Equations (43) and (44) yield 



IK^lt)! ~ 4P(,|sin(y2t/2)|/t-^ as t - oo. (45) 



The zeroth-order slender-ship approximation may be 

 expressed in the form 



KqH) = K(+(t)-^Ko-(t), (46) 



where K^(t) is given by the sum of a double integral over 

 the hull surface and a single integral along the top 

 waterline, as follows: 



Kg* = vy^' dx /^° dz exp[v2(i -Ht2)z) 



exp[ - iv2(l + t2)'/2(x± yt)](/3g -t- ygZ) 



-v^/q dx exp[-iv2(l-i-t2)'^2(x±yt)] 



Pl/[l+Pl + rl(\-x)\ (47) 



The integration in these integrals is carried over the 



positive half of the hull surface. Equations (40a) and 



(42a, b) then yield y = (/3g-fygZ)(l -x). We then have 



x±yt = l-(l±/3ot)(l-x)±yQt(l-x)z. 



By using this relation into equation (47) we may then 



obtain 



exp[iv2(l-(-t2)'^2,K±(t) = „2y^' explivh\+th^^H]TPQt)x\ 



where I*, with n = and 1, are the integrals defined as 



(- 1)"I* = v2/ j° exp[v2(l -i-t^Kl +iyoUx)z]z" dz, 

 with u defined as 



u = tAl-i-t^)'^^. (49) 



The integrals I* may be evaluated analytically with 

 the result 



Ifl* = (l-ee^)/(l-ht2)(l+iygux). 

 If = [F^(l-ee^)/(l-i-t2)(l + iy(,ux)-dee^] 



/(I-l-t2)(l+iyoUX), 



where e is the exponential function given by equation (44) 

 and e^ is the exponential function defined as 

 e^ = exp[±iv^dygt{l+t^)'^h]. (50) 



We then have 



/»olo*-yolf-/'oAi + /?^y^') = 



A^ -ee^B^/(l-i-t^)(l + iyoUx), (51) 



where the terms A^ and B^ are given by 

 A^ = Pg/(\+l^){\^iyQUX)-pl/(l+pl + yy) 



-yoFV(l-t-t2)2(l+iypUx)^ 

 B ^ =/}(,- y^d - y^F^/i I + 1^ 1 + iYq^x). 

 It may be verified that we have 



A, = Po[o-Pii^)/(i+<-^)+^¥orl/ii+Pl+Yl)]m+p^^ 



+ yoC±. (52a) 



B^/(l-l-t2)(l + iyj,ux) = p^,/{\+l^)-ygD^, (52b) 



where the terms C^ and D^ are given by 

 C ^ = [Plyln 1 + Pl){l +Pl + yl)]xhl - x2)/(l +pl + y^) 

 - [PoyfltV-hF^d -Ft2-y^V)/(l -l-t^-l-y^V) 



+ it(i -I- t^)^^HPo - 2fV(1 + 1^ +ygtV)}x] 



/{l+tH\+l^ + ylth\ (53a) 



{l+t~){l+l^ + yllV)D^ = pgYgtV + dd+t^) 



-l-F2(H-t2-y^tV)/(I -l-t^-l-y^V) 



+ it(l-(-t2)'^2[^o-y(,d-2FV(l + t^ + rotV)lx. (53b) 



Equation (50) yields 

 exp(iu2(i-i-t^)'^^(l + /3Qt)xle^ = 



exp[iv2(l+t2)i''2(i + /?jt)x], (55) 



where p^ is defined as 



/3j = pQ-dy^ = tan/3 -dtany, (56) 



as may be obtained from equations (42a, b). By using 

 equations (55), (51), (52a, b) and (44) into equation (48) we 

 may obtain 

 exp(iu2(i-i-t2)i/2]K±(,) = 



Pol(\-plih(\+t^r^l^{l;Po) + pyoU+Pi + Ylr^if(UPo) 



-(1 +/32)exp{ -v^dd +t2)}(l +lYh^{v.p^M\ +Pl) 



+ yo[J + a;/'o.C±) + exp{-v2d(l-n2)}J^(t;/3j,DJ), (57) 



where the functions l*(t;/3) and J^(t;/3,A), or more 



