SEAWAY 



75 



regular waves in model tallk^^ and indeed are so specified. 

 Barber's work also will be reviewed here. 



Once there developed an appreciation for the energy- 

 spectrum description of waves, thef)reticians t'oimd it 

 difficult to wait for a definite analytical formulation of 

 the directional parameter. 8t. Denis and Pierson (1953) 

 concluded from the work of Arthur (1949), on variability 

 of direction of wave travel, that the energy in a wave 

 spectrum travels directionwisc according to a coK-d law 

 where 8 is the deviation from the dominant direction of 

 travel of wave energy. This angular-dispersion factor 

 coupled with Neumann's version of the wave spectrum 

 resulted in an expression for the directional-energy 

 spectrum of the form 



E(cofi) = -c.-^'^^'/'^-co^-'e, {-■K/2<d<Tr/2) (136) 



This expression was used as the excitatictn in a statistical 

 approach to solution of the e(|uations of ship motions 

 (see Chapter 3, Section 3). Subse(|uently, there was 

 some belief that the angular spreading of wa\'e energy 

 was not so great and the factor cos-6 in (136) was re- 

 placed by coa^d. This was accepted tentati\'ely until 

 actual measurements suggested directional measurements 

 which are not so universal but which depend on the wave 

 freciuency (oj). The matter is by no means settled. 



8.71 Extension of the covoriance transform method. 

 Section 8.3 discusscil a fornnilation of the waxc six'cti'um 

 through the autocovariance and Fourier transform con- 

 cept. Longuet-Higgins (1957) treated the extension of 

 this method for the case of a short-crested Gaussian sea 

 surface as a function of time and space. In particular, 

 he showed that the Wiener-Khintchine relations, estab- 

 lished for a time base, could be transformed to a spatial 

 co-ordinate system without loss of generality. This is 

 to be expected, in view of the ergodic theorem. 



Consider the Fourier-series representation of the sea 

 surface as a function of time, given in (133). If this is 

 written in the general form 



?)„ = c„ cos (u„.r -|- v„ij + wj. + („) (137) 



where f„- = a„- -{- b„- and e = tan~'6„/a„, and then ex- 

 tended to a spatial cartesian co-ordinate .system, the re- 

 sulting expression given by Longuet-Higgins for a con- 

 tinuous two-dimensional spectrum is 



'?(.'•, y, n = T.C.. i-os (a„x + i\.ij + ojj + €„) (138)^» 



n 



where 



it = o)- COS d/g and v = co- sin d/g 



are the projections of the wa\'e number {k = 2ir/\) on 

 the .r and j/-axes, respecti\-ely, and co„ is a function of both 

 u„ and i\,. The amplitudes c„ are random variables such 

 that, in any element du dv it is assumed that 



" This is an al)l)revi:ited notation for tlie equivalent expression 



>) (X, y, t) = X S (■„,„ COs(i/„, X + V„ II + W„n I -F tmn) (138a) 



used by some investigators. The formulation in (138) is that of 

 Longuet-Higgins and is retained in this discussion of liis work. 



Z! ^ f"' = l'^(it,i'}dii dr 



(139) 



The (luautity E(u, v) is the two-dimensional oi- directional 

 sjjectrum of the waves. 



According to Longuet-Higgins: "the mean square 

 value of r; per unit area of the .sea surface per unit time is 

 gi\'en by 



urn 



.v,r,7'-. 



= E lc„- = ff E{u,v)dMdv (140) 



Thus the contribution to the mean energy from an ele- 

 ment du dv is proportional to E du dv. 

 We shall write 



// 



E{u, v) du dv = mm 



(141) 



and in general for the (p, g)th moment of E{u, v) aljout 

 the origin we write 



n 



u^'r''E(i(, r) du dr = m^.^ 



(142) 



The.se quantities will occur repeatedly throughout the 

 following analysis. It is a.-^sumed that they exist up to 

 all orders re(iuired. 



"The function E{ti, r) is closely I'elatetl to the correla- 

 tion function i/' (.c, //, t) defined by 



>/'(.«■, u, n 



^'f-J-S-r'"''''''''' 



urn 



.v.r.TWco 8: 



Xv (.»■' + .'■, //' + ;/, I' + Orf.f' d.y' dt' (143) 

 On suljstituting from (138) in the foregoing we find 



'/'(•(•, y, t) = I] .^ f»' '•"» ("»■'■ + >''M + ""^) 



It — 



which can be written 



\l/(x, y, t) = \ \ E(u, v) cos (ux -\- vij + o)t) du dr 



(144) 

 •so that xp is the cosine transform of E." 



li y = t = 0, the result is the spectrum of a record ob- 

 tained as a line drawn on the stereophotograph of the 

 sea surface. When the wave record is obtained from a 

 pair of stereophotographs, the elevations tj = r)(.r,;/) are 

 for a fixed instant of time (t = 0) and the spectrum can 

 be expressed as 



E {u, V) 



(27r)- J J 



t/'(,r, y, 0) cos {ux + rij) dx dij 



(145) 

 In the foregoing exposition taken from Longuet-Hig- 

 gins (1957), the directional spectrum was represented in 

 terms of ])rojections of wa\'e numbers on the x and y- 

 axes. The presentation was oriented to show that the 

 measurement of the directional spectrum EiyU, r) lends 

 itself to determination from a rectangular grid consisting 



