Internal Wave Interactions 



The solutions then take the form 



X (periodic function) 



(DIO) 



where /< is a real constant. This case rep- 

 resents the horizontal channeling of waves: 

 the negative exponential in (10) describes 

 the attenuation of wave amplitude in the 

 upward direction (z increasing), and the 



positive exponential describes the downward attenuation. (Note that this is not, 

 of course, a dissipative attenuation. It corresponds, rather, to the attenuation 

 along a lossless transmission line excited at a frequency above the cutoff value.) 



The first range in which vertical transmission is suppressed occurs when y 

 is near v^/2. For small a, the limits of this range are approximately 



2 k,(l±S) , 



(DID 



and X is a maximum at approximately the center of it, i.e., at y = kj/2. The 

 latter case is precisely the one considered in the paper. The maximum value 



of K is 



2 sin2(; 



k.f 



kjaU 

 4n 



cos 261 



(D12) 



hence, in view of Eq. (D6) and since 



a = 7 cot 6 cos 



— k J cot d cos 



(D13) 



where ^ is the angle between the x direction and the horizontal component of 

 the wavenumber vector k = (a,/3,7), we have 



Z_ = — 



8N 



kj^ (Ucos0) 



cos 2(9 



(D14) 



This expression for the minimum value of the "penetration depth" Z may be 

 compared with Eq. (7) in the paper (here cos = aj in the paper), but unfor- 

 tunately the two results are not in agreement as regards the factors depending 

 on 6 . The reason for the disagreement has not yet been fully uncovered. Note 

 that, as would be expected for obvious reasons, the penetration depth depends on 

 the mean-velocity component cos ^ in the horizontal direction of wave propa- 

 gation. 



547 



