I 

 i 





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ft? 





Figure 10. Portion of a plane wave front intersecting a keel 

 and its reflection on the surface. A coordinate axis labeled s 

 is centered on this wav«* front. 



1 



I 



1- 



The half width b was 1.6 dvn the original distribution. This gives a maximum slope 

 angle of 51.34 degrees. Since the keels were randomly wicened by Eq. 1 to simulate random 

 directional orientation of the keels, a larger value of b is appropriate. The mean widening 

 factor can be determined by integrating Eq. I from to I. A value of n-/ 2 is obtained. This 

 gives an average half width of 10.05 m, compared -vith 6.40 m without widening. The percent 

 of shadow for b - 10.05 m and converted to decibels as described above is plotted in Fig. II. 

 The ray theory values from Fig. 8 are included for comparison. The shadow covers the 

 complete surface at a grazing angle of 4.39 degrees, and the loss is infinite. At almost all angles 

 above 8 degrees this surface shadow curve lies within the ray theory data points. The smaller 

 loss of the ray theory over the shadowing curve for angles below 7 degrees may result partly 

 from scattered rays falling into the 2-degree bins in which rays are collected. However, the 

 largest difference arises from the randomness of ihe keels, which do not shadow the surface as 

 efficiently as do keels of uniform size and spacing. 



A quick correction for this effect is to assume that the shadows are randomly spaced 

 and subtract the fraction of double-shadowing from the fraction of shadowed surface, giving 

 x*/ 2, where x is the probability of a point on the surface being shadowed. This shadowed 

 fraction is in turn corrected for triple shadowing, 5.xl so on. This series is a representation for 

 the exponential function, a familiar result from probability theory. Thus the unshadowed 

 surface l-x is represented by exp (-*). The lower line in Fig. 1 1 shows this loss for a randomly 

 shadowed surface. The resulting curve seems to serve as a lower limit to the reflection loss. 



Had we used the narrower keels having a half width of 6.4 m V>e upper line would 

 have been very similai. It is about 0.22 dB less at 5 degrees. 0. 10 dB less at "> Hegrees. and 

 back up to 0.15 dB less at 20 degrees. These narrower keels fully shadow the su..cre at a 

 4.37-degree grazing angle. The shadowing strength of the keels depends very little on their 

 width, but rather on depth and spacing. 



II 



