574 CARTWRIGHT [CHAP, 15 



spectrum", though the factor pg is usually dropped for convenience, so that 

 E{k, 6) has dimensions L^. By definition E{k, 6) is essentially positive and we 

 assume, as mentioned in the introduction, that it tends to zero for large k in a, 

 suitably convergent manner. 



For some purposes it is more convenient to use the energy density with 

 respect to frequency and direction, or with respect to component wave numbers 

 u = k cos 6, v = ksin6, yielding directional energy spectra E'{a,d), E"{u,v), 

 respectively, with the relations 



E'{a, d) = {dkjdo)E{k, d), E"{u, v) = k-^E{k, d). 



E"{u, v) is most useful for definining the statistical properties of the wave 

 surfaces ^{x, y, t) (Longuet-Higgins, 1957), particularly in terms of the moments 



1*00 1*00 



rripq = uvv9E"{u, v) du dv. (10) 



J-OO J -00 



As a rule most statistical properties depend only on a small finite number of 

 these moments, regardless of whether moments of higher order exist or not. 

 We have already seen that moo represents the total "energy" per unit area, or 

 strictly, the mean square deviation of the surface height (either in time or in 

 space) from its mean level, which from (6) is obviously zero, (mio/moo) and 

 (moi/wioo) are the co-ordinates of the centroid of the spectrum, so that the 

 wave system may be roughly described as having a mean wave number k = 

 ■\/{miQ^ + moi^)lmoo propagating in the mean direction ^ = tan~i (moi/mio). 

 The second moments about the centroid, 



ju-ii = (miiWoo-wioWoi)/moo (11) 



/X02 = (wo2Woo-moi2)/moo, 



could be used to describe a sort of inertia ellipse, definining the mean square 

 spread of the spectrum. A more meaningful expression is derived by con- 

 sidering the intersection of the wave surface with a vertical plane inclined at d 

 to the a:-axis. It is easily shown (Longuet-Higgins, 1957) that the number of 

 zero-crossings of ^ per unit distance in this plane, which is proportional to the 

 root-mean-square wave number along the section, is 



[(W20 cos2 6 + 27)111 cos 6 sin ^-|-mo2 sin^ ^)/moo]'^S 



which is maximum and minimum in two perpendicular directions given by 



tan 2^p = 2mii/w2o — Wo2. (12) 



We may call the value of dp which gives the maximum the "principal direction" 

 of the waves, for obvious reasons, and the ratio of the minimum to the maximum 



- m20 + Wi02-[(W20-W02)2-f 4Wii2]'/2 

 /'^ = 



^20 + nio'i + [(m2o - Wo2)^ + 4mii2]'/2 



(13) 



