A. S. Iberall 

 [D2+a2+ j5R^^cp] V = 



V= A [1 + Cp(p + CjCp" + . . .] cos 



+ A [Djcp' + 'D^(j)"' + . . . ] sin 



+B [1 + E-qp + E,(p" + . 



+ B [Fjcp' + F^qp"' + . . .] cos 



ax + jS -^ — J (p 



R 1 



• c oo r 

 ax+ jb ^^/(Pj 



• c oo p 



It will be found, as in [10], that the solutions separate into odd and even 

 solutions, and that both cannot satisfy the boundary conditions because of the 

 irreducibility of cosh c^x. (Only sinh c^x can relax the secular determinant 

 to zero.) 



Further, the large magnitude of a^, and a^ make 



sin a, , = -i cos a, , 



■ ' 1,3 ■' 1,3 



because of the large complex magnitude of the arguments. Thus, the independent 

 even and odd solutions are proportional to each other, except for the fourth mode. 

 It is this relation which ultimately results in each of the two families of waves 

 incoming to the wall vanishing independently. 



The program has not yet been carried through completely, so that the con- 

 stants of integration for the core have not been fully related to the boundary- 

 layer constants. However, the independence of the two solutions of different 

 symmetry is clarified. 



Another task that had been neglected was the demonstration of a second 

 "stable" branch for turbulence, namely, a law of mean flow other than the g = 

 AR^Q law of laminar flow. 



This can be obtained crudely as follows. 



Roughly, the boundary- layer thickness is of the order of the limiting eddy 

 size. Our theory says 



I = h//3co^ . 



If we use the new expression for 1 - cp this had a peak in R" at R" ' = 0. 

 Whence x^ = 2N- 3/2N- i or i- !x| = 1/2N. (Laufer's data show l-x = 

 1/4N instead.) 



so that 



P/h = \/(ico^ = 1 



/3^o = 2N 



1/2N 



744 



