Peak direction is defined as the direction of maximum variance density in 

 the directional distribution associated with the peak frequency. In symbols, Q p is 

 the discrete direction at which S(f , 9 m ) is a maximum. It is interpreted as the 

 direction of the most energetic waves at the frequency containing the greatest 

 overall energy. 



Circular Moment Parameters 



Kuik, van Vledder, and Holthuijsen (1988) proposed a useful set of parameters 

 that define mean wave direction, directional spread, skewness, and kurtosis based 

 on circular moments of directional distribution functions. Though derived for di- 

 rectional distributions at individual frequencies, the definitions can be applied to 

 any directional distribution function. For the purposes of characterizing a 

 frequency-direction spectrum as a whole, the direction spectrum S(0 m ), as de- 

 fined by Equation 1 9, is used herein because it represents total wave energy in any 

 given direction arc. 



To define a directional distribution function (one that integrates to unit area) 

 from the direction spectrum, 5(0 m ) must be normalized by its own area. By 



Equation 20, this area is identically — H* B , so the appropriate directional distri- 

 bution function is 



D ( d J = -^- 5 ( m) m = 1,2 M (2 1) 



H mo v ' 



Circular moments in terms of Z)(6 m ) adapted from definitions by Kuik, van 

 Vledder, and Holthuijsen (1988) are 



cos 



(0 m - e )D(9 m )rf8 (22) 



m = l 



», = E sin(e m - e )D(e m )^e (23) 



m = \ 



m 2 = £ cos(2 m -2d )D(Q m )dB (24) 



m = l 

 M 



n 2 = Y, sin ( 20 m - 2 6 )Z>(ej</6 (25) 



m = l 



where O is the mean direction defined by requiring «, = . With this constraint, 

 Equation 23 can be solved to find 



Chapter 4 Characterizing Parameters 1 5 



