precisely I*(t;/3,v) and J^(t;P,A,v), are defined in terms of 

 the integrals 



I*(t;/3) = v^Jq exp[iv2(l+t2)'''2(lT/?t)x]x2" dx, (58a) 

 J^(t;fl,A) = v'^j^ exp[iv^(l+t^)'^2(lT/n)x)A(x)dx.(58b) 



The integrals I*(t;/3) can be evaluated analytically, 

 with the result 



lo±(t;/3) = i(l-E,)/o^, (59a) 



lf(t;/J) = -i[(l + 2iF2/o+)E^ + 2F*(l-E^)/o2^]/o^,(59b) 

 where o ^. and E ^ are defined as 



o^ = (l+t2)'^2(lT|3t), E^ = exp(iv2o^). (59c.d) 



Expressions (59a,b) for Iq (t;/?) and l|+(t;/?) are not valid 

 in the special case when we have t = 1/^, for which 

 expressions (59a, b) become 

 Io+(l/|3;/3) = i/^ l,^(l//3:)3) = vV3. (59e,0 



The integrals J ^ defined by equation (58b), where 



the amplitude function A(x) takes the form of C^ or D^ 



specified in equations (53a, b), cannot be evaluated 



analytically. These integrals were then evaluated 



numerically by dividing the integration range < x < 1 



into N segments of equal length and using piecewise 



quadratic approximations for the amplitude function A(x) 



within each segment. In this manner, we may obtain 



N 



o + J,(t;/3,A) '^Z(£^>'-'[i(A.-£^Ai^i) 

 j = l 



+ (NF^/o + )(£+-! )(Aj + I - Aj) + (2NF2/0 ^ ) 



{l+£^+i(2NF2/o^)(£^-l)}(A. + Aj^,-2A3^l^2)], (60) 



where o^ is given by equation (59c), £+ is defined as 



t^ = exp(iv^o ^ /N), (60a) 



and A:, A-^j, A. ^1^2 represent the values of the 



amplitude function A(x) at the points Xj = (j - 1)/N, Xj^ j 



= j/N, and "f-.^^/i = (j-l/2)/N, respectively. Expression 



(60) for J^(t;/3,A) is not valid in the special case when t 



= l/p, for which we have 

 N 



6NJ ,(l//3;/3,A) cy 



X A; + / 



j = l 



(60b) 



In summary, the zeroth-order slender-ship 

 approximation Kg(t) is determined by equations (46), (57), 

 (59a-f), (60), (60a,b), (42a,b) and (56). In the limiting case 

 Yq = 0, these equations yield 

 Ko(t) = 4/3o(l+t2)-3/2[(i_p2,2)(i^^2)-l 



-exp{-v2d(l+t2)}](R-il)/(l-/32t2), (61) 



where the terms R and I are given by 

 R = sin[v2(l+t2)'^V2] cos[v^(l+t2)'''2/2] 

 -pgt sin[v%t(l +t^)'''^/2] cos[v^/3Qt(l +t2)''V2], (61a) 

 I = sinV(l+t^)'^V2]-sin2[v2/3gt(l+t2)'^V2]. (61b) 



In the thin-ship limit Pq -' 0, equations (61) and (6la,b) 

 yield 



Ko(t) ~ 4/Jo(l-t-t^)-3/2[i-exp{-v2d(l-)-t2)}]. 

 sin(i/^(l -(-t2)'^V2] expl-iv^d -i-t^)'^V21 as ^^ - 0. (62) 

 The thin-ship limit (62) of the zeroth-order slender-ship 

 approximation KQ(t) may be seen to be identical to the 

 Michell thin-ship approximation (43) in the particular case 

 Yq = 0. However, the thin-ship limit (62) of the slender- 

 ship approximation (61) is not uniformly valid in the limit 

 t -• °°; indeed, equations (61) and (61a, b) yield equation 

 (62) in the limit PqI -* 0. Equation (62) yields 

 |Ko(t)| ~ 4Po|sin(v2t/2)|/t3 for I « t « IZ/Jq. (63) 



More generally, equation (61) yields 

 |Ko(Ol '^ 4/3o|R-il|/(l +Pln^ as t - °o. (64) 



Equations (61a,b) show that |R-il| ~ /3Qt|sin(v^Pot^)|/2 if 

 ^yt » I. We then have 

 |Ko(t)| ~ 2pl\sm{v%t^)\/{\ +p2),2 for , » yp^, (65) 



In the limit t-*o°, equations (57) and (59a-d) yield 

 exp(iv2t)Ko±(t) -v /3(,[(1 -py)l^(l:Po) 



+ yo(i+Po + roy^P¥^t^^'M^n+py as t - CO, 



where we have 



lo*(t;/5o) '^ i[l-exp{iv2t(i + /)gt)}]/(l + /3(,t)t as t - CO, 



If (t;/3o) ~ -' exp{iv-t(l + /3ot)}/(l +/}(,t)t as t - CO. 



We then have 



|Ko(t)| -- 2/3o|N|/(l +Pl)\l-Plt^\t^ as t - CO. (66) 



where the term N is given by 



N = (l-/3gt^)[sin(i'^t) + i cos(v^t)) 



- (1 - o2/32t2)[/3gt sin(v%t2) + i cos(v2/3(,t')], (66a) 



with o^ defined as 



o^ = (l+plmi+pl + rl). (66b) 



We may then obtain 

 |Ko(t)| ^ 4/3o|sin(v2t/2)|/(l+/32)t3 for 

 I « t « F/(/3(,)'^^ (67a) 



|Ko(t)| -v 2pl\sm(v%th\n\+pl + yy for t » \/pQ. (67b) 

 Equations (67a, b) are identical to equations (63) and (65) 

 in the limits /Jq ~* and Xo ~* ^' respectively. More 

 generally, equation (66a) yields the following upper bound 

 for the term |N| in equation (66) 

 |N|2 < (1 -pli^f + (\ +Pllh0 -o-/32t2)2 



+ 2(1+PqI)Hi +oPQiW -/3ot)(l -o/3ot)|. (68) 



Equations (67a, b) thus show that in the limit l -" <^ 

 we have |Ko(t)| -^ l/t^ for 1 « t « F/(/3o)'^^ that is for 

 moderately large values of I, and |Kg(t)| ^ l/i^ for \/pQ 

 •<3C t, that is for very large values of t. The asymptotic 

 approximations (67a, b) and (45) show that the Michell 

 approximation Kf^(l) corresponds to the thin-ship limit Pq 

 « I of the zeroth-order slender-ship approximation K„(t), 



10 



