Phillips 



Using Eq. (D6) again, we can rewrite Eq. (D14) in the form 



, ,3/2 



8 (N^ - n^) 



kj^ (U cos 0) |2n- 



(D15) 



which shows clearly that the penetration depth for y - V^/2 is least when the 

 wave frequency approaches the Vaisala frequency. Note also that z^-*™ (i.e., 

 there is a window for transmission upward or downward) when n -•N/v^ and so 

 d -» 77/4. At a given frequency the horizontal wavenumbers a and /3 correspond- 

 ing to y - ki/2 are given by 



2 (N^ - n^) 



and the limits of the channeled range of wavenumbers are given approximately 

 by this expression times l ± a. 



Channeling also occurs for small f when y is near the values 



7 = -|ki , s = 2, 3, 4 (D17) 



But the widths of these higher ranges become progressively narrower with in- 

 creasing s . Also the respective minimum penetration depths corresponding to 

 Eq. (D14) are proportional to f " ^ approximately for small f, and so they 

 greatly exceed z^ in the first range. The higher wavenumber ranges in which 

 channeling occurs appear to be much less significant, therefore, than the first 

 range discussed in the preceding paragraphs. 



REPLY TO DISCUSSION 



O. M. Phillips 



In these comments, Dr. Benjamin makes a most valuable contribution. The 

 problem is viewed here from the point of view of single scattering theory, 

 whereas the interaction problem corresponds more to multiple scattering. 

 Nevertheless, one might have expected the results to have been in qualitative 

 agreement, and the appearance of the window in Dr. Benjamin's analysis but not 

 in mine is a difference that must be resolved. 



Further work (which will be described elsewhere in due course) has indi- 

 cated that the discrepancy has arisen as a result of different specifications of 

 the current structure, which in both cases is periodic in the vertical direction. 



548 



