Dagan 
been observed previously by Salvesen (1969) is the singular beha- 
vior of the waves amplitude and wave resistance at small Froude 
numbers. If € is kept constant, and no matter how small, the 
second order wave resistance becomes unboundedly large in compa- 
rison with the first order wave resistance as F-—0. Hence the linear 
theory, as well as the second order correction, become inadequate 
at small Froude number, although both C,, and Cy> tend to zero 
as F-—0O. This effect is called subsequently ''the second small 
Froude number paradox". 
Finally, we believe that our method of computing the wave 
resistance of n urces by starting with the solution for two sour- 
ces offers a possible efficient way of attacking three-dimensional 
problems. 
III - SMALL FROUDE NUMBERS PARADOXES. 
III .1 - Introduction. 
We have seen before that the computation of the wave resis- 
tance by the thin body expansion, which is the method universally 
used at present as far as the free surface condition is concerned, 
becomes doubtful at small Froude numbers. This could be observed 
only after evaluating the second order terms. Experiments also 
support the conclusion that the linearized theory fails to predict 
correctly the wave resistance at low speeds. The aim of Chaps. III 
and IV is to elucidate this problem The same subject has been 
considered previously by Ogilvie (1968). Some of his ideas are vali- 
dated by the present study, but his solution is shown to be incomple- 
ue: 
III .2 - Solutions in the Potential Plane. 
As long as we seek solutions of two-dimensional flows it is 
more convenient to operate in the potential plane f= ¢@t+iy , as 
the plane of the independent variable, rather than the physical plane 
z=x+iy, in order to derive results of principle. The advantage 
stems from the fact that the free surface is kept at the fixed and 
known location w=0. Hence, we consider now the solution of w/(f) 
(fig. 5b) analytical in the half plane y <0 cut along |¢|<1, 
=-h+0 satisfying the following condition, equivalent to (2), (3) 
and (4) 
oh Payee oe Eger =0 ( y = 0) (32) 
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