Yih 



and substituting (6) and (7) into (2), we have, with P denoting -p'/ p, 



[p(U-c)2f']' + p[Pg-k2(U-c)2]F = 0, (9) 



which is the equation used by Miles [ 196i] and Howard [ 1961] to 

 study the stability of stratified flows. 



Miles [ 1961] assumed U to be monotonic and U and p to 

 be analytic in his studies, Howard [ 1961] was able to prove Miles' 

 theorem (on a sufficient condition for stability) and to obtain his own 

 semi-circj^e theorem without these hypotheses. But both of them 

 assumed p to be continuous, and considered the upper boundary to 

 be fixed as well as the lower one. We shall now show that the 

 theorems of Miles and Howard can be generalized to allow density 

 discontinuities. The mean velocity U (though not necessarily U') 

 will be assumed continuous. 



Let there be n surfaces of density discontinuity, and let the 

 free surface, if there is one, be the first of such surfaces. The 

 densities above and below the Jj-th surface of density discontinuity 

 will be denoted by (p )> and (pi): > respectively, and we shall define 

 (Ap)i by 



(Ap)j =(p^ -7^);, (10) 



The interfacial condition can be obtained by integrating (9) in the 

 Stieltjes sense in an arbitrarily small interval containing the discon- 

 tinuity under consideration, and is, with the accent Indicating differ- 

 entiation with respect to y, 



[p(U-c)2F']y - [p(U-c)2F']i =-gA?F, (11) 



to be applied at any surface of discontinuity. At a free surface py 

 vanishes, and (11) becomes 



(U-c)2F'=gF, (12) 



which is the free-surface condition, to be applied at y = d, d being 

 the depth. If the upper surface is fixed instead of free, the condition 

 there is 



F(d) = 0, (12a) 



The boundary condition at the bottom, where y = 0, is 



F(0) = 0, (13) 



222 



