Wu 



When co^ -^ N^ , it is expected that the nonlinear effects will become important. 



Although this is a highly idealized special case, its mathematical simplicity 

 facilitates a thorough investigation of the detailed flow features, and these fea- 

 tures may cast light on more complicated general problems. Some theoretical 

 and experimental study of this type of flow has been done by Gortler (14). 



Dispersion Relation 



We propose to investigate once more the elementary wave 



4 = f(z) exp[i(k^x+kyy-a;t)] (41) 



propagating in an unbounded medium free of wave source. The solution of f(z) 

 is 



f = e^^(Ae''^% Be-''^n. (42) 



where 



1/ 2 



k,2=k,2+k/. (43) 



Hence 



k^2 > 



according as o) '^^ ~ — . (44) 



K' --^ < / ' {0 



Thus CO- co^ separates the motion oscillatory in z from that aperiodic in z. 

 Rewriting (43) and (44), we obtain the basic dispersion relation as 



k. 



(k^2 +/32_cr2) 



(l<z>0) (45a) 



For plane waves in x-z plane, k = 0, one simply replaces k^ by k^ in these 

 formulas. This dispersion relation (45) is plotted in Fig. 1, with level lines and 

 paths of steepest ascent on the c^ -surface shown. The projections of these lines 

 on the kj.-k^ plane are shown in Fig. 2. 



For a wave oscillatory in z, of given wave number vector k = (k^,ky,k^), 

 the phase velocity c is 



558 



