which varies from to 1 like the expression given in Equation 15. Angles 

 representing the 25-, 50-, and 75-percent directions, named Q^^x ids > 

 ^50%, IDS ' ^"'i ^75%, IDS ' respectively, satisfy the relations 



1(^25%. ids) = 0.25 (21a) 



1(^50%, ids) = 0-50 (21b) 



1(^75%. ids) = 0.75 (21c) 



in analogy with Equations 16(a-c). A bulk spreading parameter derived from 

 these angles is 



^^IDS = ^25%, IDS " ^75%, IDS (22) 



and represents the angle subtending the central 50 percent of the total energy 

 in a wave field. 



153. The parameter given by Equation 22 includes the effects of any 

 frequency -dependent scattering mechanisms in that energy is summed across all 

 frequencies to get ^{&^ ■ If low- frequency waves have more shore -normal 

 directions than high-frequency waves, the total energy can be smeared across 

 the direction axis. The proper total width is given by Equation 22 if one is 

 not concerned with the detailed frequency distribution of energy. If this is 

 of concern, a second definition of bulk spreading may be more appropriate. 



154. In the second definition, the spread parameters for each frequency 

 A^„ are multiplied by the corresponding frequency spectral densities S(fj,) 

 and summed. The result is normalized by the total energy to obtain a spec- 

 trally weighted mean directional spread A^s„ . The mathematical expression 

 for this is 



TuT^ I S(fn) A^„ (23) 



61 



