Yih 



f(y) = (c - U)F(y). (29) 



In terms of f (y) , the governing equation (9) becomes 



[ c-U 



(Pf)' + P^-^ -kp - 



(c - U)^-l 



f = 0. (30) 



Equation (30) can be made dimensionless by the use of the new 

 variables 



r f ~ ^ * y TT U A c ,-,. 



f=V' P=D ' '^"d' ^=V' ^=V' ^^^^ 



where p Is a reference density and V a reference velocity. Then 

 (30) becomes, after the circumflexes are dropped, 



in which everything is now dimensionless, the accents indicate differ- 

 entiation with respect to the dimensionless y, 



a = kd (33) 



Is the dimensionless wave number, and 



N = gd/V^ (34) 



Is actually the reciprocal of the square of a Froude number. The 

 appearance of N does not necessarily signify the Importance of sur- 

 face waves, since It appears even if the upper boundary is fixed. 

 The fact that It Is associated by multiplication to p' Indicates that 

 the entire term represents the effect of gravity In a stratified fluid 

 In shear flow. 



Henceforth In this paper we shall consider rigid boundaries 

 only, for which the boundary conditions are 



f(0) = and f(l) = 0, (35a, b) 



to be Imposed on the function f In (32). 



226 



