159. Figure 9a illustrates one such case. In that figure there are 

 three maxima: (a) a small one at about 50 deg azimuth, (b) a main one at 

 about -10 deg azimuth, and (c) an intermediate one at about -60 deg azimuth. 

 By the definition given above, only the minimum marked ^^^j, between the last 

 two maxima is considered significant. The distribution to the left of 6^^^ 

 is considered one mode, and the distribution to the right of ff^^^ is con- 

 sidered another mode. The directional bounds on the first mode are 90 deg on 

 the left and 9^^^ on the right. The second mode begins at 0^^^ and extends 

 to -90 deg on its right. The number of modes in this case is two. 



160. A measure of the relative importance of each mode is the fraction 

 of total area (or energy) under the directional distribution contained in each 

 mode. To do this, it is necessary to integrate D(f^,d^) over the parts of 

 the distribution associated with each mode. Since D(f„,^^) integrates to 

 unity over its full 180-deg range, the partial areas under the modes auto- 

 matically give the fractional part of the total energy. If the distribution 

 does not go to zero in the gap between the modes, the tails of the modes 

 overlap and some assumptions must be made about their treatment. In this 

 crude first approach, the assumption is that overlapping tails have no area so 

 that integration over the limits of a mode distribution gives an estimate of 

 the (normalized) energy in the mode. At a mode boundary (i.e., B = 6^^^^ ), 

 half the area in the direction bin (of width d^ ) is assigned to each mode. 

 At the alongshore azimuths (+90 and -90 deg), the energy is very small; 

 therefore, the same boundary rule can apply there without loss of generality. 



161. The symbol P^*^' is used to represent the fractional area in the 

 k mode (counting, arbitrarily, from the left) of a directional distribution 

 at frequency fj^ . Clearly, two parameters of a mode definition are the 

 directions that bound a mode. If the indices of the discrete directions 

 bounding a mode are r and s , the notation for these directions is 



9^1^ , the direction boundary on one side of mode k at frequency fj, , and 

 ^3^^ , the direction boundary on the other side of mode k . The boundary 

 directions are either an alongshore azimuth or 9^^^ as shown in Figure 9. 

 Note that at 9^^„ in multimodal distributions, 9^^l = ^r'^n^' , i.e., the right 

 bound for one mode is the left bound for the next mode. 



162. With these definitions, an expression for the fractional area 

 (energy) in mode k is given by 



63 



