generality. The only restrictions are hardware limitations of measuring and 

 analyzing real data. 



47. To discretize frequency, the full range of frequencies of interest 

 (herein those associated with wind waves) is divided into a number N of 

 increments of size df . If fj, is the center frequency of the n*'^ incre- 

 ment , any wave having a true frequency in the range f„ - ^df < f < f „ + 2df 

 is considered to have the frequency fj, as an approximation. Direction is 

 discretized in the same way. The range of directions of interest (-90 to 



+90 deg for incident waves near a coast) is divided into a number M of small 

 arcs of size d6 . A wave with a true direction in the range d^ - 2^9 < 6 

 ^ ^m '^ 2^^ i^ assigned the approximate direction 9^ , where 9^ is the 

 center direction of the m*'*^ arc. 



48. Within this type of frequency- direction coordinate system, it is 

 relatively simple to write a formula for a complex sea with several contribut- 

 ing wave trains. As an extension of Equation 1, it takes the form 



N M 



r?(x,y,t) = I I a^ cos(27rf„t - k„x cos 9^ - k„y sin 9^ + <f>^) (5) 



n=l m=l 



where 



N = number of frequency components 

 M = number of direction components 

 a,^ = wave amplitude corresponding to frequency fj, and direction 



fjj = center frequency of the n'''^ frequency band and defined by 



f„ = fi + (n - l)df for n = 1,2,. ..,N , with 



fi = center frequency of the first frequency band (typically 

 near 0.05 Hz for wind waves) 



ffj = center frequency of the last frequency band (typically 

 near . 3 Hz for wind waves) 



kj, = wave number corresponding uniquely with f„ for a given depth 

 d through Equation 2 



9^ = center direction of the m^*^ direction arc and defined by 



e^ = 9^ + (m - l)d9 for m = 1,2, . . . ,M , with 



21 



