A discussion of the statistical methods utilized here is given in References 3 and 7. 



There is considerable evidence^ to indicate that the distribution of wave heights cor- 

 responding to any one given sea condition is of the one-parameter type known as the Rayleigh 

 distribution which is defined as 



P{x) ^l-e-''^/^ 



where E is independent of x. Thus the probability is defined by a single number* E. On the 

 other hand, when the heights of all waves experienced over a long period of time, say over 

 several years, are considered, then the evidence indicates that the logarithm of the wave 

 height is approximately normally distributed, that is, the two-parameter log-normal distribution 

 describes the situation. The log-normal distribution is defined as follows: 



(logx-fi)2 



1 ^ 



p (log x) d (log x) = — ^ e 2 0^ d (log x) 



where u is the mean value of log x and a is the standard deviation of log x. 



Reference 3 shows that these two types of distributions also describe the response of 

 the ship to the waves. For the sake of brevity, the distributions applicable to homogeneous 

 conditions of the sea, ship speed, and course will be called "short-term" distributions, 

 whereas the function which represents the distribution when the seas, ship speeds, and 

 courses are allowed to vary over a range of conditions, will be designated as "long-term" 

 distributions. 



The distribution pattern will, at a glance, give the probability of exceeding any given 

 magnitude of motion or stress. It also can be applied to the prediction of the largest magni- 

 tude to be expected in a given number of variations. For application to design for endurance 

 strength, the distribution pattern can be utilized as a load spectrum. Illustration of these 

 applications will be given in a later section. 



*E is the mean value of x . 



