stability of and Waves in Stratified Flows 



so that a^ U ^ b, we have 



> WU - a)(U - b)Q = r U^ - (a + h)\ UQ + abC Q 



= [c,^ +c.^ - (a +b)c^ +ab]yQ +ygpB|F|^ +2,g^ip|^l^' 



after using (23), This means that 



[ c^ - (a + h)/lf + c,' < [ (b - a)/2]^ (25) 



that is , the complex wave velocity c for any unstable mode must lie 

 inside the semi-circle in the upper half-plane, which has the range of 

 U for diameter. Thus Howard's semi-circle theorem is recovered. 



From (19) and noting that |W| — ^\ » we deduce that 



k^c,^ < max (U'V4 - gP) (26) 



remains valid even if there are surfaces of discontinuity in density. 

 In (26) we exclude these surfaces in the evaluation of p. It is easy 

 to see that (26) contains Miles' theorem, 



V. SUFFICIENT CONDITIONS FOR STABILITY 



Miles' theorem gives a sufficient condition for stability. But 

 it certainly does not guarantee instability if the local Richardson 

 number J(y) defined by 



J(y) = -^ (27) 



U' 



is less than 1-/4 in part of the fluid or even all of the fluid. We shall 

 sharpen Miles' sufficient condition for stability by deriving two 

 theorems which constitute, more than anything hitherto known, the 

 natural generalization of Rayleigh's theorem for the stability of a 

 homogeneous inviscid fluid. 



For the discussion in this section it is more convenient to use 

 the stream function 



L|; = f(y)e"<(''-<^*^. (28) 



Comparison with (6) and (8) shows that 



225 



