H^o = 16 ^ ! S(f„,^J df d^ (10) 



n=l ni=l 



59. Either of the two sums in Equation 10 can be replaced by the 

 corresponding single sums that define SCf^) or S(5„) . Equivalent alterna- 

 tive definitions for H^^ are, therefore, 



H^„ = 16 I S{8J d9 (11) 



in=l 



and 



H^, = 16 ^ S(fJ df (12) 



n=l 



Equation 12 is the conventional definition, arising from the more commonly 

 observed frequency spectra computed from a time series of observations at a 

 point in space. 



60. Two other common parameters are characteristic frequency and 

 characteristic direction for a given sea state. These are typically as- 

 sociated with the frequency and direction at which the most energy is con- 

 centrated, corresponding with peaks in the spectra. The symbol fp is used 

 to denote the frequency at which a spectrum has a maximum in energy density. 

 Both the frequency spectrum S(f) and the frequency-direction spectrum 

 S(f,^) usually have peak frequencies, but they may not be the same. If a 

 frequency-direction spectrum has a narrow directional spread at one frequency 

 and a broad spread at a second frequency, the peak of the frequency-direction 

 spectrum would be at the first frequency. When summed over all directions, 

 however, the total energy at the second frequency may be greater. In this 

 case, the integrated frequency spectrum would have a peak at the second 

 frequency. To distinguish the two in notation, additional subscripts are 

 added. Thus, f p fd equals frequency at the peak of the frequency-direction 

 spectrum S(f,^) ; and f jpg equals frequency at the peak of the integrated 

 frequency spectrum S(f) . The latter measure is the conventional definition 

 deduced from a time series at one point in space. 



61. In the same sense, the symbol 6 (peak direction) denotes the 

 direction at which there is a maximum in energy. Conflict can occur here 

 between peaks in the direction spectrum S(^) and the frequency-direction 

 spectrum S(f,^) . The two peak directions are distinguished by the notation 



25 



