The 25 percent and 75 percent angles would switch places, but the magnitude of 

 their difference would be unchanged. 



147. A second useful parameter gives a measure of the degree of 

 symmetry in a directional distribution. Almost all models of directional 

 distribution assume that the energy is symmetrically distributed about some 

 peak direction. If so, the peak direction ^p „ would correspond to ^50%, n • 

 There is no assurance that this is necessarily so in shallow seas, where 

 processes such as bathymetrically induced scattering, nonlinear interactions 

 among shoaling waves, and generation by irregular winds could induce sig- 

 nificant asymmetry in directional distributions of incoming waves. In the 

 simplest asymmetric distribution, the curve will be steeper on one shoulder 

 than on the other. The integrated cumulative distribution will rise rapidly 

 on this shoulder with the effect that the 50 -percent angle will be nearer the 

 quartile angle on the steep side. A simple way to characterize this is to 

 define an asymmetry parameter Aj, as 



'25%, n 



^50%,n I 

 '75%, n J 



A„ = In ^^^^ ^^^" (18) 



l "50%, n " ''75%, n J 



which is unique to a particular frequency f^ within a given frequency- 

 direction spectrum. 



148. This parameter simply compares the spreads of the two halves of 

 the central 50 percent of the energy at this frequency. If ^50%, n ^^ halfway 

 between ^25% n ^^'^ ^75% n > ^^ means the two halves have roughly equal 

 spreads as measured by the difference angles. The ratio of the differences, 

 in this case, is unity. The natural logarithm of one is zero so there is zero 

 asymmetry (by this definition) . If ^50%, n ^^ closer to ^25%, n than to 

 9y^^ n , the ratio is less than one and its logarithm will be negative. In 

 this case, the distribution rises more rapidly on the 25 -percent side so that 

 a negative Aj, indicates a rapidly rising left side and a more gradually 

 decaying right side for the angular coordinate system shown in Figure 8. A 

 positive Aj, indicates a gradually rising left side and a steeper right side. 

 Use of the logarithm makes it possible to compare distributions which are 

 asymmetric but mirror images of each other. Both will have the same magnitude 

 of Aj, , but one will be negative and the other positive. 



59 



