SECT. 5] ANALYSIS AND STATISTICS 581 



We can define the energy spectrum by 



E{cT)8a = 2iCn^ (0<a<oo) 



8a 



possessing moments 



Too 



nip = aTPE{a) da (24) 



which are assumed to exist up to any required order. It is clear that the spec- 

 trum of any scalar oscillatory quantity linearly related to ^ is simply related to 

 E{a) ; for example, the spectrum of vertical acceleration of the surface is 

 a^E{G), and, using (4), it can be seen that the spectrum of first-order pressure on 

 the sea-bed is {pg sech kh)^E{a). 



The statistical properties of i,{f) are defined by the moments of E{a) as in 

 section 4 of this Chapter, but are simpler. Values of ^t) chosen at random or at 

 equal intervals of time have a normal distribution with mean zero and variance 

 Wo, and the covariance of l^{t) with ^{t + r) is 



so that 



^{t) = C{tK{t + T) = \ E{a) COS GT da, 



1 f'^ 

 E{a) = - (/((t) cos CTT dr. 



TT jo 



(25) 



The variance of ^'{t) is m2, and of <^"{t), m^. l,'{t) is uncorrelated with either 

 (,{t) or t,"{t), but the latter pair have the covariance ( —1712). Slightly less easy to 

 l^rove is that the average number of zeros of l,{t) and of l,'{t) per second are 

 respectively 



Nq = 7r-i(m2/mo)'/^ and Ni = 7T-i(m4/m2)'/^ (26) 



iVo"! is the mean of the probability distribution of time intervals between 

 successive zero crossings of l,{t). The precise form of this distribution is, how- 

 ever, unknown, and presents great mathematical difficulties. It is of interest 

 since its shape may provide useful information about the shape of the spectrum. 

 A rough approximation is given by the probability of l,{t) being zero at both t 

 and t-{-r with gradients of opposite sign, worked out by Rice (1945). The 

 approximation is improved when it is further stipulated that l,'{t) ^ and 

 ^(^ + |t)^0, and computations of this probability for various spectra, with 

 discussion of other ajjproximations, are presented by Ehrenfeld et at. (1958). 

 Longuet-Higgins (1958) obtains a sequence of approximations based on the 

 probability that l,{t) > at ri evenly spaced instants, which gives good results, 

 and also derives the useful approximate formula 



I 



277i\^o F{T)di 



ijj{r)jilj{0) = cos 

 where 



F{t) = r p{r) dr, 



P{t) being the distribution of zero intervals and rm its median. 



