If P(H) is known, Q(H) can also be computed from the result of the follow- 

 ing derivation 



Q(H) = p(x) dx - p(x) dx 

 Jo Jo 



= P(oo) - P(H) 



= 1 - P(H) (4) 



12. Conventional statistical definitions allow other properties of a 

 wave height distribution to be found from basic probability density functions. 

 For instance, the mode, or most probable wave height, is the maximum of 



p(H) . The mean, or average wave height is the integral of the product 

 H p(H) over all H . The point here is that all statistical properties of 

 interest for a given process can be found once the probability density 

 function is known. It is this function, therefore, that is most fundamental 

 to define. 



Two -Wave Model 



13. In the analysis described below, synthetic series of points are 

 computed by summing a number of sinusoidal components. Aside from the trivial 

 case of a single sine wave (where mean, mode, RMS, and all other measures of 

 wave height are constant and equal to each other) , the simplest sum is of two 

 waves of equal amplitudes, very slightly different frequencies, and arbitrary 

 initial phases. These two wave trains will beat together, forming the 

 familiar grouping pattern where the waves are nearly in phase and then become 

 out of phase over relatively long periods. 



14. Statistical properties of this two-wave process were derived by 

 Longuet-Higgins (1952). In particular, he found the pdf to be 



P2(H) 



H < 72 H^ 



(5) 



H > 72 H^„^ 



