In equation (6), the value of the integral is equal to one by virtue of 

 the fact that X is chosen by (5). Stated another way, the probability 

 distribution function* of the K percent highest waves is found from 

 equation (1) by finding that part of the area to the right of a given 

 point on the x axis such that it equals K percent of the total area, and 

 then the truncated part is amplified 100/K times so that the area under 

 the curve will again equal one. 



The average amplitude of the one- tenth highest amplitudes is given 

 by 1.800 v'E. and the average ainplitude of the one- third highest 

 amplitudes is given by 1.416 .y/E, 



3. Wave Heights 



In a simple sine wave, doubling the amplitude gives the crest- 

 to- trough height of the wave. In an irregular wave record this is not 

 necessarily the case. A study of any wave record, as for example, 

 figure 2, shows that a succeeding trough does not necessarily go as 

 much below sea level as the crest it follows goes above sea level. 

 In a swell (or equivalently, with a narrow spectrum), the succeeding 

 trough is well correlated with the crest and the wave heights are 

 approximately twice the crest amplitude. In a sea, where the spectrum 

 is broader, this is not the case. 



The results given above for wave amplitudes then need not give 

 results applicable to crest- to- trough wave heights. Apparently, however, 

 they apply, in nnany cases, to wave heights as well. 



The results of Seiwell (1948) and Weigel (1949) in an analysis of 

 wave heights bear this out in that such values as the ratio of the 

 significant wave height to the mean wave height and the mean wave 

 height to the average of the one- tenth highest waves all agree well 

 with the theoretical values which would be obtained by doubling the 

 values given above and interpreting the results as wave heights. 



The most complete study of the problem is found in a paper by 

 Watters (1953) where the crest- to- trough heights of 109 records 

 were studied. It was found that the heights were distributed according 

 to the distribution given by equation (1). The histograms given of the 

 wave height distributions are just what would be expected from the 

 sannple size and the theory of sampling. The Chi- square test was 

 applied to 38 of the records studied, and remarkably consistent results 

 were obtained which conclusively prove that the distribution given 

 by equation (1) is valid for the wave heights studied. 



♦Hereafter this expression will be abbreviated to p.d.f. 



