A. S. Iberall 



Among all the possible waves, we have selected those systems that may persist 

 (not indefinitely, but as a sample of these waves that are not dissipated). They 

 do not represent all the waves, but they represent a potential wave system which 

 deterministically can provide ever-present fluctuations and which perhaps can 

 account for the mean dissipative losses. Namely, it is this system that the 

 pressure gradient generates and which also represents the source of drag. The 

 others are evanescent. Thus what we propose is that there is an extensive dis- 

 tribution of systems that may satisfy the equation set. We chose as representa- 

 tive of this distribution, the one in which nondecaying modes exist and all decay- 

 ing modes are zero. This is one feasible set, and in our view a "typical" one 

 which should give good "typical" measures, i.e., measures near the mean. 

 Others scatter around in a suitable phase space. These waves represent a dis- 

 persive "plane" system that is self-excited. Actually, they are not really plane, 

 but curve with changing curvature in the mean field. Here we are locating the 

 asymptotic system within the boundary layer. 



SPECTRAL RANGE ASSOCIATED WITH PROPAGATION 



As the first step, we can examine these first propagation results for con- 

 sistency with the spectrum of turbulence. 



(i) The coupled results for s and X. lead to a finite cutoff for w. Let this 

 be represented by w^. Since, by inspection, s grows with w, let it take on its 



maximum value b 2 = s 2 ^ 



= 0. Then 



^= \' -- /^v = 



20'^co„ 



7 - 1\ /, 7-1 



1 + ^1 + 1 + q + 



1 + 



y - 1 



whence 



v^ 



1 i,,y-^ 



^ 



1 . ^^V . Ii.^.^-' 



We can make use of Laufer's data [8], in particular his R^^, = 30,800 data, 

 because of its completeness. We may adapt for his experiments air flow, 

 nominally normal (20'^C) temperature. 



V = 0.150 cm^/sec 



P = 1.206 X 10"* gr/cm^ 



Ai = 1.81 X 10-4 poise 



724 



