Non-Linear Shtp Wave Theory 
uniform in the entire £ plane cut along 7=h, |E/<1 and conver- 
gent for finite e, and e, , not necessarily small. 
The ''model" problem is similar to our original flow problem, 
(57) being similar to (32), €5, €,, w and ¢ corresponding to F? 
€ , w and f, respectively, and p representing the body effect. 
Two major simplifications are present, however, in (57) : the coeffi- 
cient of dw /df is given, the equation being thus linearized, and 
equation (57), with analytical coefficients, is valid in the entire plane 
and not merely at w= 0. Due to these simplifications (57) admits 
an exact closed solution. 
Our purpose is to establish, by using the exact solution, how 
can uniform expansions of w for €, = o(1) be obtained from the 
expansion of (57). 
If we expand (57) for €, = o(1), €5= 0(1) (corresponding to 
the thin body expansion) with 
z 
ORS Mette ea tee) w, +... (60) 
we obtain for ,, w, ... equations and solutions similar to (38), 
(39), (41), (42), (44). In particular, if we let subsequently €— 0 into 
(60)% 1962" 3B. ut + 6, w+ atest ac hes wo re ws + ... We arrive 
at different asymptotic expansions in the two regions of the ¢ plane 
(fig. 7a) exactly like in (47) and (48), ''wavy'' terms being present in 
the shaded region of fig. 7. If we start with an expansion of (57) for 
€> = o(1), €, = 0(1) (corresponding to finite thickness, naive small 
F expansion) with w= w + €5 w' +... we obtain equations similar 
to (51)-(54) and solutions with no waves. Furthermore, if we let 
afterwards «,— 0, i.e. w? = €, w + (€, 2 ws +..., we obtain li- 
mits which are different from those of the preceeding expansion, in 
the shaded zone of fig. 7. Hence, our model problem leads to the 
same ''paradoxes'' as the prototype nonlinear problem. 
Now, let us consider the exact solution for w , satisfying 
(57) and (58), which can be written at once as 
Las 
