Non-Linear Shtp Wave Theory 
f 2i 
w? ety a ewe ae = Oe aire we 
hai 1 2 1 
Ae —co 
2 Bie (0 
exp (ik /F°). exp[ - =f GS Naval ath (71) 
Ff 
For F*>0 WwW, > w, + o(F? ) excepting the ''wavy wake". There, 
we obtain similarly to (67), by contour integration like in fig. 7b, 
F 1 
0 2 
We eMhodiarer 3 > exp [-i(f- in) / F°] f wo?" (s + in) 
F 
5 stih 
exp (is/F ). exp v) dv] dst+... 
oo 
(Ref > o) (72) 
where w%" and wot are given by (56). w, is obviously not 
a uniform solution. 
Only, and if only, «¢/ ee se o(1) (72) degenerates into 
(47,48). The implications of the different limits are discussed in 
the following sections. 
IV .3 - Uniform Solutions in the Physical Plane. 
It was advantageous to carry out the basic derivations in the 
potential plane. In applications it is convenient (and in three dimen- 
sions it is essential) to operate in the physical plane. It is easy to 
transfer the Gans results to the physical plane. With w(z;e,F, 
h) = 1+ W(z; e€, ie ,h) we have the following limits : 
(i) €=0(1), F°“ =0(1) (thin body, finite Froude number), 
=e Wa | eo WY. , W, » Ws satisfy equations similar to 
t= (14). Ww, = dt, i, dz is the usual thin body solution, 
Wa = df, / daz is ihe second order solution (see chap. II). 
(ii) F* = 0(1), ¢€ = 0(1) (Small Froude number, full body), 
w-=w? + F? w! +... . The complete first order term we = oe - iV 
satisfies the free-surface boundary condition 
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