SECT. 5] 



ANALYSIS AND STATISTICS 



583 



of € from 0.20 to 0.67, including some cases where 2a does not fit the distribu- 

 tion (27). 



Since the minima of ll,{t) also have the distribution (28) with reversed sign, 

 the mean value of 2a, 2a say, is given exactly by 



2a = 2U = 2mo'^^ = [27rmo(l - e2)]'-, 



Equation (27) gives 2a = (277mo)'S which is consistently too high by the factor 

 Ni/Nq. This discrepancy is to some extent reduced by ignoring waves below 

 some arbitrary size in measuring 2a (Pierson, 1954), but the exact relation 

 between the mean value so derived and the mean obtained by counting all 

 crests and troughs is not easily defined. In any case the differences become less 

 important when e <^ 1 . 



Fig. 4. The family of probability distributions of maxima (28). The case e = is a Rayleigh 

 distribution, and the case e = 1 is a Gaussian distribution. 



A very sensitive recording of (,{t) may show high-frequency ripples, which by 

 increasing W4 relative to the other moments may give a value of e only a little 

 below 1. In this case, the mean of ^m is not a good measure of mo, since it 

 contains the factor {l — e^y-^, but a better measure is the mean square ^m^, 

 which has the expectation mo (2 — e^). Equation (27) gives a'^ = 2mo, but the 

 exact relation between ^^^ a^id a^ is unknown. 



Other quantities used as proportional measures of mo'^ are the mean of one- 

 third highest and one-tenth highest wave heights. The former is often called 

 the "significant wave height", though with no real justification. If 2a is used 

 for wave heights, then the above quantities can be shown, from (27), to have 

 the expectations 4.004mo'- and 5.091wo'- respectively provided e is small. The 

 corresponding factors for t,m are very nearly half these values for e ^ 0.3, but 

 decrease slowly with increasing e. 



A similar statistic, even more easily obtained, is the largest of N wave 

 heights. The ratio of its expected value to mo-'^ increases with N but very 

 slowly. For wave heights 2a and values of N greater than about 20, this ratio 

 is very nearly, for small e, 



2'/^{\n N)y^ +2yy{h\ N)-y^, 



