Sx 3 



A similar argument is applied to 3E79y, 3r|/9x, and 9r)/9y, as 



3£ _ T o v o _9n = Y o v o 3n _ _ Y o u o 



9y 3 ' 9x 3 ' 9y 3 



% q q 



Then we obtain 



9S sec 8 ( 9g Sn A sec 9 , . fl . 



ta = ^r ("ai + ^ tan e ' = — r (u o +v o tan e) 



q„ 



3S sec 9 /3£ , 9n _ \ sec 6 , . a . 



W = ~^T W W n ' = ~^~ (v o" u o } 



q o 



It is readily shown that the dispersion relation of Equation (63) is 

 satisfied by the substitution of the above relations. 



The amplitude function of the free wave at infinite downstream is 

 expressed by the complex form 



JJ' 



A(9) - -J | | [■'.(■■on) -Pi i sec &(f>n tar, (i; | ,n",d M 



00 



Y o rr -2 



= q_ D(x,y) exp[i sec 6(?+ri tan 6)] dxdy 



The integral with respect to x, y is carried out over the plane z = 

 excluding the interior of the hull. 



33 



(82) 



