Non-Linear Shtp Wave Theory 
resistance is always positive, while in an illegitimate expansion of 
cs Nt F 5 in a truncated power series it may become negative. 
Cp isclose to Cpg, (fig. 8) at relatively high F, (i.e. small 
e2 /F), but shifted towards small FY, - The peaks of the resistan- 
ce curve of the uniform solution are, excepting the highest peak, 
much smallerthan those of the usual theory. 
The disastrous effect of using the second order approxima- 
tion of the thin body expansion in the region of small Froude numbers 
is illustrated clearly in fig. 8. 
It is worthwhile to point out that the wave resistance (81) is 
integrable only if the leading (or trailing) edge singularity is like 
w, ~(z - ih+ 1)>“ where a<1. Hence, a parabolical nose 
( a=0.5) is acceptable, while a box like shape (a= 1) is not inte- 
grable. 
IV .5 - Extension of Results to Three-dimensional Flows. 
It is easy to proceed along the same lines and to derive the 
free-surface conditions satisfied by the various uniform approxima- 
tions as F-—0O in three dimensions. With u, v, w the velocity 
components and z a vertical coordinate the exact free-surface con- 
dition, counterpart of (2), may be written as follows 
(2 z 2 
F' [uu,. tuv(v, +u, )+vv,_ + uww,_ + uww, } w =0 
x x y y x y 
(z =n) (85) 
The naive small Froude number expansion (asy.v% w) = 
(uo , vo , w? )+ FA (u! Oy hw hockii = Ee n° +... leads 
to vw , v0 ,w® asa rigid wall solution for flow past the actual 
body. With e¢« a fineness or slenderness parameter, a further expan- 
sion of (u, v, w) yields u® = 1 + eu? tan, thode? owl T= 
e (v9 , wo ) +... . The usual linearized approximations is obtained 
from (85), by expanding the velocity near the uniform flow 
MS hl le ceady. bocicrs sy (visuwx' is € (v, »W, > Mm) tees as follows 
Fou Aaowe f= 20 (z = 0) (86) 
Let now 1+V® bea uniform small Froude number appro- 
ximations. By analogy with (73), inthe limit F->0, e« =0(1) © 
satisfies the following free-surface condition 
lt2s 
