stability of and Waves in Stratified Flows 



Theorem 4 explains why for the same N, given p and U, 

 there can be many values for a on the neutral- stability curve (or 

 curves), which has been observed by Miles [ 1963] for a special "p 

 and a special U. 



We can sharpen Theorems 4 and 5 by defining M to be the 

 maximum of B/^ in (y , y ) for a = 0, Then if (51) holds and 



^^{n+lfj!^^ (52) 



(y^-y,) 



the words "at least" in Theorenn 4 can be replaced by the word 

 "exactly. " 



We note that the analytic ity of p and U is needed only near 

 y., and that, as a consequence of Theorem 5, a layer of homogeneous 

 fluid containing a point of zero U" and adjoining a stratified layer 

 with uniformly large J(y) is always unstable. 



VII. NON- SINGULAR MODES 



It remains to study neutral waves with a (real) c outside of 

 the range of U, whose minimum and maximum will continue to be 

 denoted by a and b. We assume p and U to be continuous, and 

 that their derivatives as appear in (32) exist. Then if m and M 

 retain their definitions as given by (51) and (52) , except that c = a- e, 

 we have 



Theorem 6. Under the assumptions on y and U stated above, if 

 (51) holds there are at least n modes with c = a - e , a = a] (i =" 

 i , 2 , . . . , n) , and a\ increasing with i. For the i-th mode there are 

 at least n- i internal zeros. If (52) holds in addition, then there 

 are exactly n such modes , the i-th of which has exactly n-i in- 

 ternal zeros. If rpU')' is negative, then n can only increase as 

 the arbitrary positive constant e decreases. 



The proof of this theorem is by a straightforward application of the 

 first comparison theorem of Sturin. Similarly, if m and M are 

 defined by (51) and (52), except that c = b +e, where e is an arbi- 

 trary positive constant, we have 



Theorem 7. Under the assumptions on "p and U stated above, if 

 (51) holds there are at least n modes with c = b + e, a = aj (i= " 

 l,2,...,n), and a\ increasing with i. For the i-th mode there 

 are at least n-i internal zeros. If (52) holds in addition, then there 

 are exactly n such modes , the i-th of which has exactly n-i in- 

 ternal zeros. If CprJ')' is negative, then n can only increase as 

 € decreases. 



233 



