Dagan 
validity of the usual linearized solutions and to improve them, when 
necessary. This way it was hoped that a better agreement between 
theory and experiment could be achieved. 
The present work is dedicated mainly to the influence of the 
nonlinearity of the free-surface conditions on the wave resistance. 
Since at this stage we are interested in elucidating problems of prin- 
ciple and basic concepts, we have carried out the derivations for 
two-dimensional flows. Two-dimensional solutions are obtained much 
easier than the three-dimensional ones due to the use of the powerful 
tool of analytical functions. They permit to find in a simple way quick 
answers for problems which in three dimensions need a tedious nume- 
rical treatment. It is realized, however, that the final conclusions 
about the applicability of the results derived here to flow past ships 
could be drawn only after their extension to three dimensions. We 
consider, nevertheless, at each stage of the present study, the impli- 
cations of the results to associated ship problems. 
Il - THIN BODY EXPANSION. 
Il .1 - General. 
We consider an inviscid two-dimensional flow past a submer- 
ged body (fig. 5a). Let z' = x'+ iy' be a complex variable, f' = ¢' +iV’ 
the complex potential and w' = u' - iv' = df'/dz' the complex velocity. 
We limit our considerations to a symmetrical body parallel to the 
unperturbed free surface : hj is its submergence depth, 2L' its 
length and 2T' the maximum thickness. With U' the velocity of 
uniform flow far upstream, we make variables dimensionless as 
follows : 
tee Ay Oy , y= m/e , zaz! f L! ) w=u-iv=w/U’ ynen/L! 
fo 64+iV= £'/0'L" Sent ae 
oma spl tay L’, F:U' Agu)’ 
F2U'/(2 gies (1) 
n' being the free surface elevation above y'=0. 
The exact boundary conditions satisfied by w(z) » which is 
analytical in the flow domain y ¢ n(x), given here for convenience 
of reference, are as follows 
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