Dagan 
where the integrals are carried out below ReA<1l , ImX =h 
and Rev <.l., Imv =b)m the * }°and .v planes, réespecttver 
We can use now the series (59) to rewrite (61) as follows 
9 
‘tig 0 0 0 1 
ame exp(-i ¢/ ea) Aaa taky b 4 ee te er ee exp(id/€,) 
€>5 “oO 
ae : 0 0 l l = 
exp furs shee ne, co Hat...) alan 
ast 
valid for finite €; , €5 . We are now ina position to expand (62) 
for small e, and/or € > . The detailed analysis may be found in 
Dagan (1972b). Herewith, the main results : 
(i) the limit €, = 0(1) , €5 = 0(1) of (62) yields the same 
results at first order, €, w, , as the solution obtained by expanding 
(57) if, and only if, €, / €, = o(1). This last condition stems from 
the existence of the ratio e, / €, in the last exponential of (62); 
(ii) the limit € = 0(1) , €, = 0(1) of (62), wa , does not 
coincide with that obtained by the naive expansion of (57). The uni- 
form solution differs from the naive solution in the ''wavy wake" and 
does comprise ''waves''. Moreover, to obtain a first order complete 
solution we have to retain in the last exponential of (62) all the writ- 
ten terms, upto ( eS , in particular €, u,. The "far waves" are 
obtained by contour integration (fig. 7b) as follows 
w? = ts exp(-i r/o f [oP + POX) W(A) |. explin/e,) 
uniform 
nN 
cp ks [ [os i wv) | dy } dd (Rela) (63) 
f 
Again, it is emphasized that in the last exponential €, uw contributes 
at O0(1) because of the division with €5 . Going in reverse, the 
differential equation which yields the uniform first order solution, 
obtained by the appropriate expansion of (57), is 
de /ag + (i/e,) (tH + 6m) wo = (i/ €)( oo + Po HY) (64) 
1716 
