582 



CARTWRIGHT 



[chap. 15 



The heights of the waves themselves as measured by l^{t) are often measured 

 as a quick and obvious way of estimating the total wave energy, mo = moo- 

 Opinions differ, however, as to the most suitable definition of a "wave height". 

 One may consider the total wave height, or difference in height between a crest 

 and the preceding trough, or, alternatively, the heights of maxima above or 

 minima below the mean level ^ = 0. Examples of both types are shown in Fig. 3. 

 The former has intuitive appeal, and it can be shown (Longuet-Higgins, 1952) 

 that when E{g) is fairly concentrated about a single frequency, so that ^t) has 

 the appearance of a pure sine wave with slowly varying amplitude, the total 

 wave heights 2a have the "Rayleigh distribution" 



p{a) = (a/wo) e-''2a-/mo. 



(27) 



A3 A4 



Bl B2 B3 B4 



Fig. 3. A trace of an actual wave record showing some selected typical values of ^,;j, height 

 of maximum (Al-4, single arrows), and 2a, total wave height (B 1-4, double arrows). 

 Note that A2 is negative, and that for B2 and B4, 2a ^^m- In some conventions for 

 measviring total wave height, B2 and B4 would be ignored. 



If it is not assumed that the spectrum is narrow, then it seems to be almost 

 impossible to derive a theoretical expression for p{a), though (27) fits many 

 ocean-wave records fairly well (Watters, 1953). On the other hand, by consider- 

 ing the joint normal distribution of ^t), t,'{t) and !l,"{t), the exact distribution 

 of the heights of maxima ^m can be derived for any shape of spectrum. In the 

 notation of Cartwright and Longuet-Higgins (1956), it is 



p{-q) = (277)-'4ee-'/^''V + (i_e2)>/.^e-y.^2 r 



va-(')'-/e 



,-i/.^2 



e-y^^ dx 



(28) 



where 



-q = ^m/Wo'^S e2 = (moW4 - W22)/WoW4. 



The parameter e is a measure of the spectral width relative to its mean 

 frequency, and is most easily obtained from the relation 



No^INi^ = l-e2. 



(29) 



Curves of ^^(-17) for a range of values of e from to 1 (its entire range) are shown 

 in Fig. 4 ; the lower limit e = corresponds to a very narrow spectrum, and gives 

 p{r}) = rje~y-'^ (rj'^O), (17 < 0), again a Rayleigh distribution. Note that in 

 general p{r]) admits negative values of i,m, of proportion [| — 1(1 — e^)'/^] to the 

 total, while 2a is essentially positive. The distribution (28) has been successfully 

 applied (Cartwright and Longuet-Higgins, 1956) to wave records with values 



