Pierson 



this paper the function S(co,d) is given by Eq. (6) for -7t/2 < d < tt/2 and by zero 

 otherwise. 



f(^-^) =i 



1 + l0.50 + 0.82e ° cos 2i9+0.032e ° cos 



(6) 



One should note that the Fourier series representation of S(w, (9) as in Eq. (2) 

 would require many more terms to fulfill the zero otherwise condition that was 

 assumed in the above expression. 



FORECASTING DIRECTIONAL WAVE SPECTRA 



At present, my colleagues and I are attempting to predict the directional 

 wave spectrum, S(aj, i9), given the winds over the North Atlantic Ocean. If ^{u>,6 

 is the directional wave spectrum, if one has the directions, 0, 77/6, 77/3, 77/2, 

 277/3, 577/6, 77, and so on to 277 with respect to north as zero, and if one has cer- 

 tain frequencies, fj, f 2, ... , fjg, 

 diet 180 numbers, one of which would be, for example. 





df . (7) 



Stated another way, the directional spectrum is described by the variance con- 

 tributions to fifteen frequency ranges for each of twelve direction intervals at 

 each point. 



Based on some theory, and some empiricism — considerations too numerous 

 to detail here* — we have started with observed winds over the North Atlantic 

 ocean for December 9 to 17, 1955, December 11 to 27, 1959, and November 17 

 to 30, 1961. These winds have been described at each grid point of interest in 

 the problem every six hours for each of the above periods. The winds are then 

 used to predict these 180 numbers to define the directional spectrum at each 

 grid point. No adjustments are made in the wave spectra and in the forecasting 

 procedure for the entire period of the forecast. 



The output consists of numbers that describe the directional spectrum as 

 defined above at each grid point. From them, we can get the frequency spec- 

 trum by summing over direction and the significant height by summing these 

 sums over frequency, taking the square root of the sum and multiplying it by 4. 

 (The spectra discussed above are all in terms of variance, and the total volume 

 under S(co,e) equals the variance of the wave motion.) 



The first indication of a good forecast is in the verification of the significant 

 height. It tells one that the area under S(co) as predicted agrees favorably with 

 the estimate of this area as obtained from an ocean wave record obtained as a 

 time history at a point. 



''See, for example, Pierson and Tick (1964). 



430 



