Yih 



covered by Miles' criterion? What, in fact, is the 

 natural generalization of Rayleigh's theorem on the 

 sufficient condition for stability of a homogeneous 

 fluid in shear flow? 



(ii) Are there some sufficient conditions for instability? 



(iii) Miles [ 1963] has shown that the wave number at neutral 

 stability can be multi- valued for the same flow, in an 

 actual calculation for a special density distribution and 

 a special velocity distribution. Is there an explanation 

 for this , even if not completely general? 



(iv) Do internal waves with a wave velocity outside the 



range of the velocity of flow exist? How many modes 

 are there? What is the character of each mode? 



In this paper the questions posed above will be answered in as 

 general a way as possible. By "general" I mean "without numerical 

 computation." Although special calculations for special flows, 

 involving the use of computers, are important because they often 

 give us insight into and understanding of the subject, and sometimes 

 are of practical interest, results obtained in a general way are often 

 more useful. The question naturally arises: Can general results be 

 continually improved and sharpened, albeit with increasing cost in 

 labor, but without the use of computers? The answer to this question 

 necessarily reveals the attitude of the respondent more than anything 

 else. My answer to it is in the affirmative, and the results contained 

 in this paper, aside from whatever interest or merit they may have 

 for those cultivating the subject, are given to substantiate my faith. 



In addition, some straightforward extensions of Miles' theorem 

 mentioned In (I) above s and of Howard's seml-clrcle theorem [ 1961] , 

 are made to make these theorems applicable to fluids with discon- 

 tinuities In addition to continuous stratification In density. 



II. DIFFERENTIAL SYSTEM GOVERNING STABILITY 



If U and 1p denote the velocity (In the x-dlrectlon) and the 

 density, respectively, of the primary flow In the absence of distur- 

 bances , and u and v denote the components of the perturbation In 

 velocity In the directions of Increasing x and y, the linearized 

 equations of motion are 



?(Ut +Uu^ + U'v) = - P^, (1) 



"p(Vj + Uv^) = - Py - gp, (2) 



In which subscripts Indicate partlzil differentiation, t denotes time. 



220 



