SECT. 5] ANALYSIS AND STATISTICS 575 



the "long-crestedness" of the wave system. If y2 = o we have m2oWo2 = Wii2, 

 and clearly the waves have infinitely long crests propagating in the direction 

 dp (or TT-\-dp), while if y2= i the waves are very short-crested, in fact isotropic, 

 the spectrum having circular symmetry about the origin. For sea waves, of 

 course, y^ always lies well inside these two limits. If the spectrum is fairly 

 concentrated about its centroid, it can be shown that y2<^ 1 and approximates 

 to the mean square angular deviation of the spectrum about Op. 

 The expression 



'm2o cos2 6 + 2mii cos 9 sin 6 + mo2 sin^ & 

 fX20 cos2 ^ + 2/xii cos 6 sin d + [xo2 sin^ d 



(14) 



can be shown (Longuet-Higgins, 1957, p. 373) to be a measure of the mean 

 number of waves in a "group" in the direction 6, most significant of course 



when 6 = dp. 



By partial differentiation of the series (6) we see that m^o and mo2 are also 

 the mean square component surface slopes, d^/dx, dt,jdy, while m\\ is their 

 CO variance. That is, 



/an 2 /an 2 idi di 



m2o = (^j , mo2 = [jy) , mn = (^ ^ 



where the bars represent mean values over time or space. If, however, we 

 expand the series for ^{d^jdx), say, equal arguments 0w occur only in the form 

 cos (f)n sin (f)n, which have zero mean values. Thus the surface is uncorrelated 

 with its slopes, and similarly with its first time derivative. In general the 

 variances and covariances of any pair of spatial derivatives of ^ are either 

 zero or of the form ± nipq, where p + q is even, and variances and covariances of 

 measures involving time derivatives of ^ are zero or of the form + mpq^^\ where 

 p + q is odd and the suffix r refers to moments of the function a^E"{u, v). It is 

 important to note that odd-order moments must involve time derivatives, 

 without which it is impossible to distinguish between E"{u, v) and E"{ — u, —v) 

 or between E{k, 6) and E{k, d + ir). 



In virtue of the random phases, €n, oi the components of the series for ^ and 

 formally similar series giving its space and time derivatives, any group of such 

 quantities has a joint normal probability distribution, expressed entirely in 

 terms of the variances and covariances between the quantities. For example, 



, . . . , 1 ( ii^ W0262-2mii^26 + W20M ,,.. 

 P(6,l2.6)- (,^)3,^^^,,^^,, exp|-^^^ ^^ 1, (15) 



where 



Az = W20W02-Wii2, 



and a great many other such distributions could be written down similarly. 



