Non-Linear Shtp Wave Theory 
ul (¢ —-00) (33) 
Here, the variables are made dimensionless with respect to U' and 
Ll’ (fig. 5b) and h is defined as h! /L’ 
The physical plane is mapped on f with the aid of 
df 
Z= hs (34) 
which leads to an unwieldy integral equation replacing the boundary 
condition (5). We shall see, however, that in different approximations 
the body boundary condition becomes quite simple. 
We consider now two basic types of perturbation expansions 
of w(f; «, F , h) aimed to linearize (32) : 
(i) the thin body expansion, considered in Chap. II, 
2 
mit; <«, © .,h)= 1+ cw. (i: F, wht ew, (f; F, h)+... (35) 
1 
for *"e=‘o())8 TF°="0(4 ) and 
(ii) the naive small Froude expansion 
0 2 
wets .€ yok new. (ipoe sh) Bowe (ie re. ui) os (36) 
fares = o(l), e=,0(1): 
Our aim to study the solutions obtained for different limits 
c= 0,  F— 0, 
III .3 - The Thin Body Expansion. 
The thin body expansion in the potential plane yields results 
similar to those obtained in the physical plane. The mapping (34) 
becomes 
z=f+t ez, + ae +E &) (37) 
where 
z,=-fw,as, ree - fiw, - (w,)° f2 Vs dé 
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