584 CARTWRIGHT [CHAP. 15 



where y is Euler's constant 0.5772. The same expression halved, and with 

 iV(l — e2)'/2 (i.e. approximately half the number of zeros) put in place of N, 

 applies to the maximum value of ^r«- Cartwright (1958) shows that the standard 

 error of the estimate is approximately 0.64/ln N times its expectation, and also 

 works out formulae for the second and third highest waves (which have smaller 

 standard errors) and the reduction in the effective value of N when successive 

 wave heights are highly correlated, as in the case of a very narrow spectrum. 



7. Spectral Measurement 



A. Variability of Spectral Estimates 



In the preceding sections it has been tacitly assumed that the energy spec- 

 trum can be evaluated to any required accuracy provided proper measurements 

 are made. In practice we can only make statistical estimates of the spectrum 

 the expectations of which are close to E{k, 6) or E{g), but which have a degree 

 of variability depending on the quantity of data used. This variability was first 

 seriously considered by Tukey (1949). We shall now briefly discuss the main 

 criteria of variability, referring principally to the case of a single variable with 

 spectrum E{a). Estimates of directional energy spectra have similar properties 

 but are more complicated, and depend on the method used. Goodman (1957) 

 derives useful results for the variability of cross-spectra. 



Consider first an estimate of the total spectral energy mo, derived from a 

 record t,{t) of duration T. 



E{T)total = ^ J^ CHt) dt. 



This quantity has expectation mo, and it can be shown (Rice, 1945; Tucker, 

 1957) that it has a "coeflicient of variation", C.V., defined as (Standard errors 

 expectation) 2 given by 



C.V. = 2, 



r E^a) da 



Tmo^ (30) 



provided T is long enough to contain more than about ten waves. If E{a) is 

 constant over a band of width aw and zero elsewhere, (30) gives C.V. = 277/crM,T, 

 showing that the narrower the spectrum the longer duration of recording is 

 required to obtain a specified level of variability. Further, since the standard 

 error decreases only with T~'/2, the duration has to be greatly increased to 

 improve matters considerably. For other spectral shapes, (30) gives less simple 

 results, but it is convenient to express C.V. in general as 'l-rrjaeT, where o-g is an 

 "equivalent spectral width", and is very roughly proportional to the standard 

 width (moW2 — mi2)'/2/mo. 



Exactly similar considerations apply when only a filtered portion of the 



