Yih 



In fact even the existence of special singular neutral modes for which 

 c equals the maximum or minimum of U does not imply the existence 

 of contiguous unstable modes, as a consequence of the semi-circle 

 theorem of Howard. Hence we need not be concerned with these 

 special border cases. In demonstrating the non-existence of unstable 

 modes it is sufficient to demonstrate the non-existence of singular 

 neutral modes with a < c < b, where a is the minimum and b the 

 maximum of U. 



Miles [ 1961, p. 507] has shown that singular neutral modes 

 are impossible for monotonic U if J{y) > i/4 everywhere. In his 

 demonstration he actually showed that a singular neutral mode with 

 a J(y-) > 1/4 at the place y = y^ where U = c is impossible. Hence 



we need only consider the case J(y-) — l/4 in our search for the non- 

 existence of singular neutral modes, 

 of (32) is 



For J(y-) = l/4» one solution 



f.= (y- 



yJ w, 



(38) 



where 



= 1 + A(y - y^) + 



(39) 



with 



A = 



■(1 +j)l£Ur -V: +Y(ln-^)'l , (y=4) 



pU' 



(40) 



provided U' does not vanish at y = y^ . [We shall consider mono- 

 tonic U only. Hence this restriction on U' does not affect our 

 results in this paper.] The other solution is found by assuming it 

 to be of the form f|h, substituting it into (32), and_solving^ for h. 

 The result, after division by a constant (which is p^. or p at y^) , 

 is 



f2=f, In(y-y^) - [ 2A + (In •^)J] (y - y^)'/^ 1 +B(y-y^) +...], (41) 



where B is a constant. Now the Reynolds stress defined by 



T = - p uv, (42) 



where the bar over uv means time or space average, can be ex" 

 pressed in terms of f as 



0' * ZaCjt 



|— ^(f'f)je ' 



(43) 



228 



