Dagan 
It is also worthwhile to point out that only for h #0 (sub- 
merged body) is the amplitude of the free waves decaying exponential- 
ly for F—0, Otherwise, for h=0 the decay may be algebraic at 
best. 
We are going to determine the origin of the small Froude 
number paradoxes and to derive uniform solutions for the wave 
resistance as F = o0(l)tand €=.0(1) or ©«= 0(1). 
IV - WAVE RESISTANCE AT LOW SPEEDS. 
IV .1 - The Model Problem. 
Like in other nonlinear problems of hydrodynamics we seek 
a ''model'' problem which has the same features as the basic nonli- 
near problem, but can be solved exactly. We define our auxiliary 
problem as follows : determine the complex function w({, €, eam h) 
of the complex variable ¢= &§ +iX, analytical in the whole plane 
cut along X=h, <1 (fig. 7a), satisfying everywhere the differen- 
tial equation 
ie [V¥O(S5e, 6, ni) Sewn b (Eye, en) (57) 
iy poh 
subject to the condition 
w 70 ( §+-0c2,Xx <h) (58) 
o0 and p are given functions, holomorphic in the entire ¢ plane 
excepting the slit [é| <1" X=h and Ol 7 i be as | f |e More- 
over, @ and p are bounded along the slit and 1+ 0 does not vanish 
there. To simplify matters we assume that o and p admit expan- 
sions in power series in e¢, and €, of the type 
w(ebis 83 soe bd =i Daalees nalecp bad beach) 
k=1 jz0 
LL GRR ENL IN egy (28) gn ORI cRE MS, 
= 0 
aE te ene egy 
dae k=0 
ly /s.(Uorhom:), = Wvabudea beet, Yo webideedior eg» tn) 
Jed 
ea yes) (eg nl Ur sm) (59) 
jy=O k=1 
