TM WO. 377 



The teim B is an integration constant o Substituting (A"16) into (A-15) and 

 integrating; 





>Jlc<^05SLt -hJl s/A/ Jit 



-Jl^^-hJl 



1 ^ -JJd-fe 



(A-17) 



or simplifying with trigonometric relationships: 



U»= -r- - r ^ ■ - CCsC-^t-H) -^ 8 iS . (A-18) 



CA') (sO 



Equation (a-12) has been used in (A=18) to define; 



< - ARCTAW -T; = ARCTAW Jl Tr. (A-19) 



As with the solution (A-IO) for response to a step function, solution (A-18) 

 contains a steady oscillatory ccaiponent term (A' ) and a time variable solution 

 term (B')o The angle P< can be thought of as the phase shift angle between 

 the response function Oi) and the driving function ^^r^^^ ■^^ » I't is clear 

 from (A=19) that the phase angle o< increases with £in increasing time constant 

 Tj,e Likewise^ for a system with a constant T^, the phase angle o<, increases 

 as the frequency of the driving function. Alsoj, the amplitude of the instru- 

 ment response decreases as SL increases o 



In chapter II a test was made of the response of OMDUM II to a vertical 

 oscillation^ The period of oscillation was about OJJ second. The phase angle 

 o< was fotmd to be about 12°o Substituting these values into equation (A-=19) 

 written as 



'^'^ _J2- (A-20) 



results in T^.*^ 2h milliseconds » This value for T^. is some-vrtiat lower than the 

 value obtained for OMDUM III by the method described in (A-1)« 



Relationship Between Instrument Output and Driving Motion Spectra — The 

 measurement in ocean waves using the ducted meters provides a time series record 

 of the angular velocity vector of each impellero From the time variable magni- 

 tude and direction of this vector one can infer, on the basis of steady flow 

 and response time calibrations, an estimate of the Eulerian particle velocity 

 at the position of the meter system* From this velocity data one can estimate 

 averages, auto-covariance, covariance^ and their related spectral functions. 

 The accuracy of the statistical description of the actual motions is based on 

 the assumption that the statistics of the impeller motion can be identified with 

 statistics of ambient random motions driving the impellers o A simple analysis 

 can be made of the criteria by wMch one can justify the correspondence of the 

 statistics of the angislar velocity vector with the particle velocity vector. 



A-4 



