Lee 



where the w-'s are defined In Eq. (61) 



5. 3 Outgoing Waves at [x[ = cd 

 Equation (11) shows that 



Y(x,t) = -•i|^x{x,Y(x,t),t)+i($x+^y)l. 



If we substitute the expansions given in Eqs. (17) and (18) into this 

 equation and equate the terms of the same order and time harmonic, 

 we find that 



Yi(x) = j^9Ji(x,0) for i=l,2, (67) 



\ ( x^ . 1 2 2 1 



"^i + Z^^^ = g {j^^'i^i + Z^^'^) " "Z^^i^iy " 4 ^•i'ix '•"^iy)/ ^°^ i=l,2, (68) 



Y5(x) = j ^^-i^ V'5(x,0) - ^W,^>^, + <P^<P,,) 



- Ti^X^Zx ""^lyV' ^^^^ 



Y6(-)=J^^K<-'0> +^(^.^2y + <PE^.y) 



- Zg (^Ix^Zx + ^ly*P2y) 



c 



(70) 

 (71) 



i=l 



If we let |x| -♦ oo (or X. — ^ oo for a = or - rt) only the 

 pulsating sources contribute non- vanishing values (see the expression 

 for the first-order potentials in Appendix B, and for <p^ and <p^ in 

 Lee [1968]). Thus we find that 



/ rv\ ~ /^ "^"^i u / £ e''^ cos px J , , Kjy t^ \| 



^j(x,0) ~ - Qje lim i\ _ ^^ dp + JTre cos KjXJ| q 



I x|— *oo *^0 '^ ' ^ 



j(Kilxl-qi) , , 



= - JTrQje for I = 1,2.3,4, (72) 



where the Qj's are the source strengths, the qj's the phase re- 



926 



