TM No. 377 



enabling it to produce artificial correlations in the wave motion data. In 

 this discussion, it is again necessary to resort to the simple sinusoidal models 

 with reference to the wave perturbations. 



The simple expression for a covariance associated with the vertical and 

 horizontal velocity components of a two-dimensional progressive wave having a 

 small phase shift &A between the oscillations is given in equations (V-39) 

 and (V-1+0) as: 



V = Au, cos <rt , , . . 



(V-U5) 



lo' = Auj sin (r*+A+) 



A u and A w are the amplitudes, ff" - 277" T , and A<p is the phase angle 

 shift. (The plot of the vector having u' and w* orthogonal components as a 

 function of 0~"fc appears in figure V-U3D.) It has been shown that for ^<t5sO 

 ^rO 5 and for A$ %o , 



jq = A±A? s/n/^4?. (v-u6) 



For small angles of A^ (less than 10°) the following approximation may 

 be written: 



U'tj = AuAw A^ . (V_U7) 



The covariance in equation (V-U7) is only an artifice to help visualize the 

 parameters controlling correlation properties. This formulation may be asso- 

 ciated with either the original source motions or the output functions of the 

 measuring instrument. A knowledge of these motions and of the instrument 

 mechanism is therefore necessary in order to define the causes of the covariance; 

 i.e., whether they are spurious or "real". These causes, if due to the instrument, 

 can be ascertained by examining the sources of possible instrument distortion 

 and their effect on the parameters A u , A w and 4&<$> . 



Direction Calibration Differences - Instrument bias or undesirable contri- 

 butions to the covariance may be produced by a difference of forward and backward 

 flow calibration in one or both of the impeller systems. Suppose that the slope 

 of the calibration curve (see figure 11-21 ) for a single -impeller is different 

 for each flow direction. If this impeller were coupled 'orthogonally with an 

 impeller having no calibration anomaly, and then placed in a regime having pure 

 sine and cosine motional components, the response would be given by: 



1^9 



