Dagan and Tulin 



the total source flux is zero. Similarly Ng is zero at both the origin 

 and infinity. In fact the first order solution gives the exact values 

 of N at infinity and at the stagnation point, the higher order approxi- 

 mations correcting only the free- surface shape between these two 

 anchor points. The above expansion is consistent and hopefully 

 uniforrnly convergent. It differs from that suggested by Ogilvle 

 [ 1968] who has kept terms of different order in the same equation 

 in order to obtain waves far behind a submerged body. 



The solution of the first order approximation for the box-like 

 shape body (Fig. 2a) is obtained in terms of the auxiliary variable 

 t, as 



w, 



= (f^)"' ^, = ^A ("" 



where the mapping of the linearized Z plane (Fig. 2b) onto t, 

 (Fig. 2c) is given by 



Z =i(C^ - if^ +iin [(t,^ - 1)'^^ - t,] (11) 



Hence , by Eq. (6) we have 



For the second order approximation (Eq. (7)) we get 



^ ^^e_ii)il (e< - 1, IX = 0) (12) 



^2 given along the t, real axis (Eqs. (8) and (12)), leads by a 

 Cauchy integral to 



1 i 2 ./etiV^^in[-g-(e^-ir] 



u 



TtT 



(iL\y nd^^jf^j^m^ (13) 



Ug and Ng as functions of ^, are easily found from Eqs. 

 (13) and (9) (for details see Dagan and Tulin [ 1969]). The shape of 

 the free- surface at second order is given in Fig. 3. As expected, 

 the profile becomes steep as Fr^ increases. 



612 



