Dagan 
via (79) for ¢€—40. Hence, the usual linearized approximation used 
in ship resistance theory may be extended in the range of small 
Froude numbers only if e°/F = gT'/U'e is small. To a € 
fixed and, to jet F*-+ 0 is tantamount to write in (80) ¢ ~21 ire 
1 - 2ie - /¥* which is obviously illegitimate if « /F* is. not 
small. For this reason the second order approximation Zie 7 /F* 
may become large compared to unity and the expansion of (79) may 
diverge. For « /F* = 0(1) (81) leads to 
7A) 2 
0 pip: ica is /F 
Al bin = ane T(s)e ds (83) 
i.e. the usual complex amplitude of linearized waves. 
IV .4 - Illustration of Results : Wave Resistance of a Biconvex 
Parabolical Body. 
To illustrate the main features of the thin body small Froude 
number solution (79, 80, 81), presented here for the first time, we 
have computed the wave resistance for a parabolical body (fig. 8) 
with the thickness distribution t=1-s* (|s|< 1), r= -2s. The 
linearized rigid wall solution in this case is expressed by 
= = ot oa = faz - = [(z - ih)” - 1) 2m 
zZ°= ih +1 i Saye 2 iibit ft 
Say Apia we -1]h ae aed. (84) 
and w? = at? /dz. By using (81) we have computed Cp (82) by a 
simple Simpson integration, for e=0.05 (Dagan, 1972b). Cp le 
as function of F, =U'/(2gL') is represented in fig. 8. On the same 
figure we have sspreuenend Cy lone based on the usual linearized 
approximation (83). In addition’ we have represented C je= Cy at 
eC,2 based on an expansion up to e© Of (82), i, es on an iMegine 
mate expansion of the exponential in (80). 
Our small Froude number thin body solution (81) differs 
from the usual one (83) with two respects : while (83) may be regard- 
ed as a summation of elementary waves e~'‘'~° generated by 
each element of thickness dt = 7ds, in (81) the elementary waves 
e ~i(f-s)}F2 have phase and amplitudes depending on > ina complex 
manner. In particular waves are generated by the parallel part of 
the body (7=0, t= const) and the amplitude changes if the direc- 
tion of motion of an assymetrical body is reversed. The wave 
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