Non-Linear Shtp Wave Theory 
the different terms resulting from the substitution of the real and 
imaginary parts of df, jk /dz into the last term of (11). The complex 
potential of the flow generated by p- , for instance, is (Wehausen 
Jk 
and Laitone, 1965) 
oo 
= i ~ zZ Z- Ss 
f. ceca Mi! (s¢%, cee Mth, Ft) w ( ) ds (22) 
jk nF’ J jk : Fr’ 
fi and ie , corresponding to pP:: and or of (20), respective- 
ly, are obtained from fjk by letting x) — x, 
Carrying out the detailed computations (see Dagan, 1972a) 
yields for the streamfunction far downstream 
2 2 2 
- a sae 
(x, 0) =Ime ix /F {teres [Er 2 git /F )> 438° }4 
Vik 
k 
where 
2 2 
s 2 2 
gs netic Nea PES es ee et eee 
7 2 4h 
F 
2 2 
B® = 2e7/F a4 2eh/Fy 
ieee daly fgets ea Sic a am 
ere 2 aes 5 Teale Sena 
ste he 2 Fa jk ico 2 
, h 
be ea 
1703 
