Davis and Oates 



an expression which is easily derived from the fact that the mean square wave 

 elevation is unchanged by the coordinate transformation. The orbital velocity 

 expressions are transformed in a similar fashion, and, along with Eq. (34), 

 formed the basis for the simulation. An example of the effect of the transfor- 

 mation on wave elevation spectra is given in Fig. 11. 



Simulation of the random seaway 



The basic method used for simulating the random seaway is common in 

 analogue computer practice, and involves the use of suitable linear filters to 

 shape the output of a random noise generator to obtain signals with the desired 

 power spectra. The simulation was done entirely in moving coordinates, and 

 thus all spectra were functions of frequency of encounter. In addition, the sim- 

 ulation was done for constant craft velocity only, since varying velocity would 

 require filters with changing characteristic frequencies and consequent extrav- 

 agant use of analogue computer components. Head, following and beam seas 

 were simulated for both the quarter and full scale hydrofoil craft for speeds of 

 25 and 50 knots, respectively. 



The starting point for the simulation was the vertical velocity spectrum 

 since, in sea coordinates at least, wave elevation is obtained therefrom by an 

 integration rather than a differentiation. In moving coordinates, wave elevation 

 is obtained from vertical velocity by a "transformed" integration, the charac- 

 teristics of which can be derived by considering the frequency response function 

 of an integrator as a function of frequency of encounter. The frequency re- 

 sponse of an integrator in sea or fixed coordinates is 



Kj^,) = JL (37) 



where o^ is the sea frequency. Using (35), the frequency transformation given 

 in terms of frequency of encounter is 



1 ± }/l -^xo;' (38) 



2V/g 



Substituting (38) in (37) will give the frequency response of a "transformed" in- 

 tegrator, thus 



l'(j-') = —. , '^^^ , . (39) 



JT [l ± VI - (4V/g) o,'\ 



It can be seen that I'(jai') has the same 90° phase lag for all frequencies as the 

 ordinary integrator, but that the magnitude is quite complicated. While (39) is 

 obviously non-realizable in the strict sense, it can be approximated over a 

 range of frequencies by a combination of minimum and non- minimum phase net- 

 works. The procedure was to first approximate the magnitude without regard to 

 phase with a combination of first and second order filters, and then correct the 



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