Kaplan 



■I 



' (i,i) 



dco $ (co) (46) 



2 



is equal to Zo", , i.e. twice the variance of the ordinates on the cor- 

 responding time-history curve. Under the assumption that the 

 seaway is a Gaussian or normal stochastic process which is exciting 

 a linear system (in this case, the barge), the set of responses of 

 the system will in turn represent a Gaussian stochastic process. 

 The probability of an ordinate of a particular response lying between 

 two values is given by the definite integral of the Gaussian proba- 

 bility density between those two limits, and will be a function of the 

 variance (Tj . Thus, Ej or ctj may be used to estimate the proba- 

 bility of the occurrence of instantaneous values in any range of 

 interest, for any given barge motion, including infrequently-occurring 

 large or near-maximum values. Characteristics of the motion time 

 history may be obtained in terms of the quantity Ej by relating the 

 behavior of the envelope of the record (interpreted as the instantane- 

 ous amplitude of the time history curve) to this quantity. Such 

 relations are based on assumed narrow-band behavior of the energy 

 spectrum, and yield expressions for the mean amplitude of oscilla- 

 tion (half the distance between the trough and crest of an oscillation), 

 the mean of the highest l/3 of such amplitudes (known as the signifi- 

 cant amplitude), and other related statistical parameters of interest 

 for a specified sea condition. In particular the relations for average 

 pitch amplitude and significant pitch amplitude are 



Qay = 0.88 ^fEB 



(47) 



V =1.4lV% 



VI. DISCUSSION OF RESULTS 



Computations of the amplitudes and phases of the six separate 

 motions of the moored barge for the complete range of possible 

 headings were carried out for wave lengths varying from 100 feet to 

 800 feet, which covers the range of periods significant for ship 

 motion in an operational environnnent up to Sea State 5, Solutions to 

 the equations were obtained for the complex response operators 

 (both amplitude and phase) of the various motions relative to the wave. 

 Representative solutions for a particular wave length, for both the 

 longitudinal and lateral motion amplitudes, as functions of the heading 

 angle P are shown in Figs. 2 and 3. From this data the r espons e 

 amplitude operators, as functions of frequency (since co = V2irg/\ ), 

 are obtained and representative curves are presented in Figs. 4 and 

 5, Application of the techn 4ues of spectral superposition theory [ i] 

 results In spectral energy density values for particular barge motloni^ 

 In Sea State 5 (as an example), and these values are Indicated In 

 Figs. 6 and 7, as illustrations of some of the results. 



1038 



