(f) = U <J> Q + ^ + (j> w (109) 



We should remember that the order changes by e when differentiating <J) or 

 (j) 1 by y or z. Picking up the time- independent part, we obtain 



d \ . „ 3 \ 3 *0 . 3 d \(^0\ 2 , 3 +0 d \ , g ^0 



+ 2 v^v^ + t — jtMtHH + 



~ 2 3x3y 3y 2 „ 2 \ 3y / 3x „ 2 I7 2 3z 

 3x ' 3y 3y U 



+ 0(e 3 ) = (110) 



2 

 Then the term of the lowest order is (g/U ) 3(J> n /3z which has the order of 



e. Therefore, the free surface condition for the steady potential of the 



lowest order is 



8z 



- = at z = (111) 



that is the condition of double body flow. We cannot proceed to the next 

 step without handling the nonlinear terms in the free surface condition. 

 The periodical part of the free surface condition can be written as 



/3 +II 3\ 2 , + ^l + 9 TT 3 *0/3 +TT 3 \ d h 



^ + ^ J_ (^0 ) 2 *1 + ,2 (jV) 2 !\ 

 y 3y 3y \ 3y / 3y \3y / a 2 



2 



+ 2 U 



3x3y UJ , uy yuy , VJ w , 3y . 



2\ .2 



-'^(fc«fc)cw^(£*i(£) ^ 



+ 0(6e 2 ) = (112) 



44 



