upstream and 



p ■+ - jcjpu 1 xe J + p 1 e J + p ls (35) 



p -*- - ja)pu 2 xe J + p 2 e J + p 2s (36) 



downstream. Here 0) is the frequency, j , the imaginary unit of time, 



u-. and u„, the fluctuating velocities in Figure 37b, and x, the distance 



along the chord. The first term may be seen to be the pressure gradient 



due to the "mass oscillations" of the up and downstream flows. The terms 



p.. and p„ are the steady pressures across the cascade in the absence of 

 Is 2s 



motion, and p and p are the additional complex fluctuating pressures 

 caused by the presence of the cascade. 



These additional pressures are what are needed to account for the 

 insertion of the cascade between the up and downstream flow regions. It 

 turns out that these additional pressures can be simply related to the 

 unsteady axial fluctuations by a linear equation of the form 



(37) 



where z is a 2 x 2 matrix with complex coefficients. This matrix, we can 

 now see, is the transfer function for the cascade. Before discussing some 

 of the terms of this transfer function, we need to mention some of the 

 difficulties in finding practical evaluations of the implied formal 

 solution. 



Even in the linearized free streamline theory, we need to be concerned 

 about the physical significance of the cavity model. It has become 

 acceptable to think of steady flow cavity models as having an ultimate 

 "wake" to account for the momentum deficiency of the cavitating drag. 

 Unsteady cavitation models have not, however, received as much attention. 



77 



