Non-Linear Shtp Wave Theory 
Z 
where terms up to ( €,) have been retained in 1/(1 +0), the 
coefficient of (59). 
(iii) the limit ¢, = o0(1), €2 = o(1) of (62) is not defined un- 
less we specify the order of ¢,/ ¢,. For ¢ /¢€, =0(1) we obtain 
again at first order a solution with a ''wavy'' term, the latter having 
the expression 
i€ 
. 0 l Z 0 
wW = Ww ans i " 
uniform ‘i 1, uniform Bias o exp(-if /¢ BI J pf ) 
bey a 2 
. exp(id/e,). exp[ fuer) ar] ar + 0(« fen 
2 €5 1 il 2 
(Re f= oo ) (65) 
This solution differs from that obtained by taking the limits «— 0 
firstand e,—+0O afterwards in (57). Moreover (65) is obtained from 
the solution of the differential equation, derived from (57), 
Or. ds (GO 
ga er ue (66) 
IMS) 
0 : 
Pie ge) w = iy? eS) py 
(iv) the limit = o(1), e,= (1), & / €, = 0(1) yields by the expansion 
of 
ie A ie d 
exp fi) a] =l1+ — f we Hide in (65) 
Bin i aint 
the usual linearized approximation 
0 ea 0 
@.= = exp (-if /e ) fe (A) exp (it /e€_) dA (67) 
1 €> 2 4 1 2 
satisfying the differential equation 
0 
0 ; 
dw, +df +(i/ej)o , 
= Miidse gh ny (68) 
a 
