Dagan 
F* Aa + aes y 25 ae aoe 7 yg (rhac s + i Vv °7 o) zi 
2 
a cad Gilg a i Pie ae ie NG i PE 5 Se ME eS 
yy: xZ y Z 
(z = 0) (87) 
@° may be represented along the actual body surface by a distribu- 
tion of singularities derived 22 2S the rigid wall solution. The rigid 
wall solution is asymptotic to ® as F390 excepting a ''wavy wake" 
which this time is generated by rays emanating from the body image 
across z=0 towards x—oatan angle -§ (arbitrarily small) 
with the horizontal plane. Todetermine ® , representing waves 
over a stream of variable velocity, is a very difficult task. 
The simpler approximation of thin (slender) body € = o(1) 
small Froude number F = o(1), and finite beam (and draft) Froude 
number €/F*=0(1), is obtained from (87) like (75) 
: 0 
0 1-2€e wy 0 
Bpe_ pol sree ater brass 0 (z= 0) (88) 
F 
a may be represented by the source distribution of the rigid wall 
solution on the body skeleton (central plane, or axis). (88) is the 
extension of the usual linearized free-surface equation (86) (which is 
the basis of computation of ship wave resistance via Michell integral) 
into the range of small Froude numbers, where (86) becomes inva- 
lid. The solution of a is the object of future studies. 
IV .6 - Discussion of Results and Conclusions. 
The two small Froude number paradoxes have been explained 
with the aid of our model problem. 
It was shown that the naive small Froude number expansion 
does not yield a uniform solution, the region of nonuniformity being 
the ''wavy wake". 
The elucidation of the second paradox has led to the impor- 
tant conclusion that the usual thin body first and higher order appro- 
ximations are valid only. for large thickness Froude numbers. For 
moderate values a new first order approximation has been derived : 
it results from taking in the free-surface condition a basic variable 
1724 
