Mooring and Positioning of Vehicles in a Seaway 



The power spectral density of the square-law device output 

 is then 



,00 



7 C 2 



S aM =-\ Rr(T)e''"''dT 



—2 



= 4(r^) 6(a)) + - R^(T)e'"*"' dr (134) 



where 6(u)) is the delta function. The last term on the right in 

 Eq. (134) can be evaluated by using the definitions in Eq, (132), so 

 that 



|rR^,(T)e-'-^dT =4pS,(co') dco' rR,(T)e-*^'^-'^'^" dr 

 ^ J.oo ^ ^-00 J-oo 



= \ Sx(co')Sj,(a) - CO*) dto» (135) 



J-oo 



which leads to the final result 



— 2 P°° 



S^2(w) = 4(r2) 6(w) + \ Sx(to')Sx(w-co') dco'. (136) 



*^-00 



The power spectral density of the square-law device output, which is 

 proportional to the drift force, contains the delta function term (that 

 represents the non-zero mean value of the force) and a convolution 

 integral of the input spectral density whose value will depend on the 

 nature of the particular input. The square law detector output is 

 obtained by operating on the output of the square law device with an 

 ideal low pass zonal filter that filters out completely the high fre- 

 quency part of its input, thereby leaving only the low frequency part 

 representing the envelope. 



A particular application to illustrate the results of applying 

 this analysis is given for the case of a vessel moored in a seaway 

 such that an irregular beam sea is present. The vessel chosen for 

 this illustration is the same CUSS I moored barge treated previously, 

 and the sea condition is represented by a Newmann spectrum cor- 

 responding to a 24 kt. wind (upper Sea State 5). A mathematical 

 representation of a filter circuit whose amplitude characteristics 

 are approximately the same as the square root of the Newmann 

 spectrum formula (Eq. (43)) was derived, programmed on an 

 analog computer with a white noise generator Input, and produced 

 an output that represented a continuous time history of the surface 

 waves with that desired spectrum, r.m.s. value etc. A simplified 

 constant coefficient second order differential equation for ship heave 



1073 



