Dagan 
IV .2 - Application of Results to the Nonlinear Problem (Potential 
Plane). 
Due to the similarity between (32) and (57) the results obtain- 
ed in the model problem can be extended to the hydrodynamical pro- 
blem at once (for details see Dagan, 1972b). With w=1+W the 
uniform solution, the key to obtaining first order uniform approxima- 
tions W° (for F2 =o0(1), ¢=0(1)), €W, (for e=o0(1), F*=0(1)) 
and e¢ wr €=0(l1), F* =0(1), e/F* = 0(1)) is to expand the 
coefficient (w)° w of dw/df in (32) ina naive small Froude number 
expansion and to retain the appropriate number of terms. By doing 
that we obtain the following uniform asymptotic approximations for 
W : 
(i) e= 0o(1), F? =0(1), i.e. ¢€/F* = 0(1) (thin body, finite 
length Froude number, large thickness Froude number (U'@ /gT'), 
W=eW, +... . W, coincides with (44), and the usual thin body 
approximation is, therefore, uniform. 
(ii) F? = 0(1), €=0(1) (Small Froude number, finite thick- 
ness), W = w? +... . W® satisfies the free-surface boundary con- 
dition (similar to 64) 
0 
z dW 
im {iF* [(u°)? + F°(u")* (3 tiv’) +... - w 
( ¥ = 0) (69) 
and along the body 1 + wo =w, where w® (51,52) is the naive 
small F solution. w® is not an uniform solution in the ''wavy 
wake". The solution of W° subject to (69) is very difficult. 
(iii) F2 =0(1), €= o0(1); ‘“e/F2 = 0(1)*(small Froude 
number, thin body, thickness Froude number ule / gl' of order 
one), W =e Wey 
Wy; satisfies the free-surface condition (similar to 66) 
0, dw, 0 
Im Lar’ (nae a Met Wy = 0 (% = 0) (70) 
where u2 = Re w? is the naive linearized small F solution (56). 
Also, along the body skeleton Sp (fig. 7) a = w, . By analytical 
continuation and integration by parts Ww, has been found in a close 
form as follows 
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