Ogilvie 



be found. Also the variance of U(t) and U(t) can be found and designated as -//^ 



and 02- 



Then the joint probability density function u and U can be given by 



Given this equation, the probability density function for f can be derived. It is 

 given by 



kja j\ /kja f 



^lV'2 Jo 



P(f)dt = "^ e Me ^ ' +e ^ ' df . 



277- xA/^ 



The time hist9ry of f is not given. The time history could be obtained by 

 generating u and u as functions of time given the free surface T7(t) in a manner 

 quite similar to that of some of the other papers in this symposium. The non- 

 linear operation corresponding to U(t) |U(t) | could then be carried out in the 

 time domain and the time history of the force on the pile could be constructed. 

 This has in fact been done by Reid (1958). In this analysis, however, the only 

 thing desired is knowledge about the probability density structure of f(t). This 

 density structure can be obtained simply by reading off equally spaced values of 

 this force and plotting the histogram of the values that are read. 



This probability density function as given by Eq. (33) has been compared 

 with values obtained directly from measurements of the forces on an actual 

 pile. Though there appear to be a number of parameters involved in Eq. (33) 

 there are really only two, the second moment and the fourth moment, since 

 P(f) is an even function. 



When these two parameters are determined from the data consisting of a 

 twenty-minute long record of the fluctuations in this force, and used to con- 

 struct P(f), the resulting probability density function agrees remarkably well 

 with the observations. The probability density is not Gaussian; in fact, it pre- 

 dicts probabilities about ten times those of the normal distribution, three stand- 

 ard deviations from the origin. 



It is also interesting to comment that the computation of the bi-spectrum 

 of f(t) would not have been particularly revealing because the bi-spectrum 

 resolves the third moment of a distribution into frequency pairs. The third 

 moment of this distribution is essentially zero. 



Work described by Tick (1963) has shown that it is possible to take a linear 

 Gaussian model for the long crested seaway and construct the model that would 

 satisfy the equations of motion in the Eulerian frame of reference to second 

 order. One result is that there is a correction to the frequency spectrum. A 

 second result is that the profile of the waves changes as a function of time at 

 a fixed point. The crests become higher and sharper and the troughs become 



94 



