Mooting and Positioning of Vehioles in a Seaway 



If the wave system, and the resulting relative heave motion, 

 are represented by the sum of a larger number of terms of different 

 frequencies, with these frequencies having only small increments 

 relative to a single reference frequency (as a result of the narrow 

 band assumption), the contribution to the drift force value in that 

 case can be shown to be given by an expression that is also identified 

 as the square of the envelope of the total signal. This identification 

 and interpretation of this type of expression for the drift force can 

 generally be extended to the case of an arbitrary input, including a 

 random input which is assumed to be made up of a combination of 

 different frequencies within a narrow band. Thus a simulation 

 technique in the time domain for this term requires determination 

 of the envelope of the input signal (relative heave motion), squaring 

 this quantity, and applying the appropriate constants to produce the 

 required time history signal. 



The drift force has been shown to be a nonlinear function of 

 the wave amplitude, and in a random sea it is a slowly varying 

 function of time, where this slow variation is considered relative to 

 the wave frequencies and the linear wave- induced forces. Since 

 the wave surface elevation and all linear terms derived from it are 

 assumed to be Gaussian random processes, the drift force is known 

 to be non-Gaussian in regard to its probability density. In order to 

 obtain further characterization of the properties of the suction force, 

 it would be useful to determine the probability distribution and 

 spectral properties of this force. The accomplishment of this task 

 will be aided by the simplified Interpretation of the drift force that 

 was presented above. 



Considering the drift force as the square of the envelope of a 

 Gaussian random process, certain Information Is available concern- 

 ing the probability density of this type of function. A square-law 

 detector produces an output proportional to the square of the envelope 

 of the Input, If the Input Is a narrow band Gaussian random process 

 [ 20] which Is the assumption used In the present analysis. For a 

 particular Input Into such a square-law detector, the probability 

 density function of the output (denoted as w) Is given by 



1 -w/F(w) 

 p(w)=^^e , w>0 (127) 



where E(w) Is the mean value of the square-law detector output. 

 On that basis, the probability density for the drift force In a random 

 sea can be represented by 



p(Fy) = J-e ^ ^ Fy> (128) 



where F Is the mean drift force, and hence the probability dlstrl- 



1071 



