56. Where the full frequency-direction spectrum is known, one can 

 compute another useful entity known as the direction spectrum. A direction 

 spectrum is a measure of the total wave energy from each direction where the 

 total is found by adding the contributions from each frequency band for a 

 given direction arc. The direction spectrum is denoted by 5(6^) and is 

 computed by 



S(ej = I S(f^,9J df (9) 



n=l 



57. The direction spectrum is a relatively new entity, primarily 

 because there are so few observations of the complete frequency-direction 

 spectrum that it has not been computed. It is extremely useful for charac- 

 terizing the bulk properties of an energy distribution where contributions 

 from all frequencies, not just a particular frequency, are considered. The 

 symbol on the left of Equation 9 is often abbreviated as, simply, S(^) . 

 Since it is derived from an integration of S(f,^), it can be called the 

 Integrated direction spectrum (IDS). 



Some Fundamental Bulk Parameters 



58. Total energy of a wind wave field is proportional (by the factor 

 pg ) to the volume under the frequency-direction spectrum. This is an 

 important parameter with which to characterize any wave field. It is commonly 

 expressed in the form of a characteristic wave height E^^ . This parameter 



is derived from the fact that the volume under a frequency- direction spectrum 

 (sometimes called the zeroth moment, hence the subscripts "mo") is identically 

 the variance of sea surface displacement, having dimensions of length squared. 

 The square root of this is the standard deviation of sea surface displacement. 

 Under the assumption of a Rayleigh distribution of wave heights, the average 

 height of the highest one -third waves is nearly four times this standard dev- 

 iation. Whether or not the Rayleigh distribution approximates the wave field 

 everywhere, four times the sea surface standard deviation has come to be 

 regarded as a characteristic parameter of a wave field. Formally, H^^ is 

 derived as 



24 



