6-^ = center direction of the first direction arc (typically 

 near -90 deg for the coordinate system defined above) 



^[^ = center direction of the last direction arc (typically 

 near +90 deg for the current coordinate system) 



(^j^ = initial phase of the wave train associated with the n*^^ 

 frequency and m*'*' direction 



49. With this type of bookkeeping, a real sea can be characterized by a 

 finite (but arbitrarily large) number of wave components. With N frequen- 

 cies and M directions, there is a total of M x N components. A simple way 

 to archive or plot this information would be to store the two one -dimensional 

 arrays of frequency f„ and direction 6^ and the two two-dimensional arrays 

 of amplitude a,^ and initial phase 4>^^ . This procedure would completely 

 define the sea surface and all other wave field properties to the accuracy of 

 linear wave theory and to the resolution of the discretized frequencies and 

 directions . 



50. However, the interest here is in wave energy. Because of the 

 discretized frequencies and directions, the energy of a wave train at frequen- 

 cy fj, and direction 6^ is simply proportional to 



E'(fn.<'m) = 2^, (6) 



as in Equation 4 for a single component wave train. When multiplied by (the 

 nearly constant) pg , Equation 6 gives the (ultimately desired) mean wave 

 energy per unit wave crest for this wave component. 



51. Though expressed at a particular frequency and a particular 

 direction, Equation 6 actually represents the energy of all waves that have 

 frequencies in the small frequency band df centered on f^ and directions 

 in the small arc d9 centered at ^„ . In results from a variety of experi- 

 ments, the frequencies and directions can be (and usually are) discretized in 

 different-sized frequency bands and direction arcs. If various step sizes are 

 used in data from the same experiment, i.e., from the same wave field, a 

 variety of energies will be obtained by Equation 6 because more or less of the 

 frequency and direction domain is included in each energy estimate. This 

 effect is eliminated if the energy measure of Equation 6 is divided by the 

 product of df and dd . The result tells how densely energy is concentrated 



22 



