Dagan and Tulin 



From Eq. (31) we obtain 



n 



t^t) i=0 b 



which is the central part of our analysis. 



The expression of the second order solution, satisfying 

 Eqs . (22), (23), and (32), was found to be 



wa(U=-^ + 0(i„U+^-^2 (33) 



i ' 



Summarizing the results for the outer expansion, we have, 

 with the estimate t = 0(6^) 



( C)'^^ j.i + 3/2 Z-nt, „i*-V2 



+ eO(^'/2in;) (34) 



1/2- 



~i-.[ai + 2a(i) ]+c^-^.|iin^ 



i t 



IT Ll r\^ 



1/2 



(35) 



e. and e^. being again constants related to C|. , c^' 



The velocity has the familiar square root singularity at first 

 order and a source singularity at second order. The free-surface 

 is continuous and attached to the bottom at first order, while at 

 second order it rises at infinity. The eigensolutions of the problem, 

 which represent in fact the linearized solutions of a free-surface 

 flow past a flat horizontal plate, as well as the flow details near the 

 bow will be subsequently determined with the aid of an inner solution. 

 It is worthwhile to nnention here that only at second order are the 

 details of the adopted model (i.e. the jet) manifested in the solution. 

 Any other model attached to the bow will produce an identical first 

 order solution. 



618 



