of the transducer for this circumstance), it was found that the convection velocity was about 

 0.8 Uq. It was expected that the convection velocity would decrease with increasing values 

 of frequency as a consequence of the quadripole nature of the right hand terra in Equation [1], 

 The measurement procedure was not sufficiently precise to reveal sucli a relationship even 

 though there is reasonable certainty of its existence. 



It can be seen that the spatial pattern of the pressure fluctuations is not convected 

 downstream as though it is frozen. If this were so, the real part of the normalized cross 

 spectral density can be shown by a simple computation, for the case of a filter with a rectan- 

 gular frequency response, to be 



Sin 



f^x \ \ 0.8f/„ / f2nfQX 



— cos 



'12 \O.SU^j I2n.\{x\ \ 0.8 f/^ 



0.8 (.' 



where the first factor on the right is an effect due to the finite bandwidth A/. Since, in these 

 measurements a constant 10 percent bandwidth was used, this factor is negligibly different 

 from unity. 



The departure, for the real part of the data of Figure 5, from cos 2 tt /^ a;/ 0. 8 {/q is a 

 measure of the development of an uncorrelated component in the pressure fluctuations during 

 its travel from the upstream to the downstream measurement point. 



A measure for the uncorrelated component can be formulated mathematically. The 

 downstream pressure fluctuation P2^^) ^^^ ^^ resolved into three components whose cross 

 spectral densities with the upstream pressure fluctuation are, respectively, real, imaginary, 

 and zero. That is, 



where P,„= U.^, P.n = iVi2^ ^jy = 0- The vanishim.; of tlia sp^^ctral density is equivalent 

 to saying that the correlation for p. (t) and y{t) is zero. This resolution can be affected using 

 the definition for cross spectral density. The spectral densities (or a(t), j3{t), and y{t) are 



P - P -IP |2/p 



10 



