where I means the imaginary part is taken. Then the solution of the 

 boundary condition of Equation (76) is given by 



•■■Jj 



E(Cn') G(S,n,C;£',n') dS'dn' (80) 



The asymptotic form far downstream, i.e., E, -> °°, is written in the form 



it/2 

 -it/ 2 



~ ^ E(£ , ,n I )d5 , dTi , I m exp[C sec 2 9+i sec QiK~V+ tan 9(n-n')}] 



sec 2 6 d6 (81) 



Thus, the phase function of the elementary wave defined by Equation (59) is 



S = Yn sec (5 +r l tan ) 



Now let us show that the phase function satisfies the dispersion relation 

 given by Equation (63) . There are relations 



d* u o 3$ v o 9 ^ 



j)£, _3¥ / 3($,y) ^0 



8$ an / 3(5,n) 3 

 7 q o 



/ 3($,y) _ _ T o f 



/ 3(C.Tl)-- q 6 h J- 



~3 



ii _ 9^ / jKjJp Y o f 8q o 

 8f " 3n / 3(C,n) " „6 U ) 3n 



~3 -1 

 Since it can be shown that 3q„/3r| = 0(Y n ), we have, after deleting higher 



order terms, 



32 



