Understanding and Prediction of Ship Motions 



as the sea can sometimes be described as a stationary random process, such an 

 assumption can lead to a description of a real sea if the angular distribution is 

 properly chosen. Without question, such a description can represent the short- 

 crestedness of the sea. The particular distribution of energy as a function of 

 angle will vary greatly with sea conditions, and it is not clear at present if there 

 is a standard distribution which will lead to generally useful results in connec- 

 tion with ship motions predictions. 



Our knowledge of the hydrodynamics of ship motions is such that we are 

 well-advised to limit our attention to the first of the two choices above, although 

 it is unrealistic. Stated bluntly, the fact is that we have far to go on the simpler 

 problem, and we cannot hope to understand ship motions in multi- directional 

 seas until we first understand what happens in artificially-produced uni- 

 directional seas. This statement need not apply if we are content to obtain fre- 

 quency response fiinctions strictly by experiment. But the principle purpose of 

 this paper is to consider the prospects for entirely analytical predictions of ship 

 motions. With such a goal in mind, we must accept that we cannot solve all of 

 our problems at once. Therefore I shall restrict myself generally to long- 

 crested seas, recognizing that a broader outlook is desirable and will ultimately 

 be necessary. 



In calculating the energy spectrum from a given wave height record, one 

 effectively discards the information which relates to relative phases of the var- 

 ious component waves. The energy spectrum gives us information only about 

 about the amplitude of the components. From the point of view of probability 

 theory, all wave height records which yield the same energy spectrum are 

 equivalent.* Then, if one wants a general representation of the surface eleva- 

 tion corresponding to a particular energy spectrum, one must allow complete 

 ambiguity in the relative phases of the frequency components. For the long- 

 crested sea, St. Denis and Pierson (1953) proposed the representation: 



^(x,t) = COS [Kx- oJt- e(aj)] /U(^^J)T^"d^ , (1) 



where 



^(x, t) = surface elevation at position x, time t, 

 [^(oj)] 2 = energy spectrum of ^(x, t), a function of frequency, w, 



g = acceleration of gravity, and 



e(co) = a random variable, with equal probability of realizing any value 

 between and 27r. t 



*That is, they are all members of an ensemble which is characterized by a sin- 

 gle energy spectrum. We assume not only that the processes are stationary but 

 also that an ergodic hypothesis is valid. 

 'A more precise definition is that p[aj<e(aj)<a2] - (a^- a^)/2TT. 



