Wu 



k = ± 





1 + 



;(N2-a;2) 



0(s.2)l 



Hence, (a) for w < n, the two branch points are in the first and third quadrant of 

 the k -plane; (b) for w > n, they lie very close to the imaginary k-axis; and they 

 are symmetric with respect to k = in both cases. The appropriate branch cut 

 for these two cases may be introduced as shown in Fig. 6. Therefore in the 

 limit as t -» +C0, we obtain the following solutions after applying (77) and Fourier 

 inversion. 



(I) w < n: 



G(x,z,t) = ^ e^^"''^* I e''"' F(k,z,(}f ) dk 



J_00 



477 



J- CO 



2„.2, .2 



zl//3^-M^k 



dk 



V/S^-M^k^ 



where M^ is defined in (51a). With the branch cut so chosen and the integral 

 path indented as shown in Fig. 6, it follows that 



G(x,z,t) = --e 



/3z- iojt 



Hy'(|yjTTi^) (x^ 



> M2z2) 



(78) 



= -2;^^'^-"^«o|V^^T- 



(x^ < M^z^) 



(79) 



.( 1) 



where Hg denotes the Hankel function, and Kq the modified Bessel function of 

 the second kind. Equation (78) shows that there is an outgoing wave in the region 

 x^ > M^ z^, its phase being constant along x^- m^ z^ - const, at any instant. These 

 salient features have already been predicted by our previous geometric construc- 

 tion of the waves. In the region x^ < M^ z^, however, our previous approach was 

 only strong enough to predict the argument of the functional dependence. 



(w<N) 



Itnk 



-/9/M 



(o) > N) 



Imk 



fi/» 



-0/1* 



Fig, 6 - Branch cuts of m for 

 CO < Nand <x) > N 



568 



