Figure 9 Angular scattering functions 

 for a set similar to keel set A, except 

 that the keel width is reduced by half 



SURFACE SHADOWING 



The curves of Fig. « appear to be a measure of the unshadowed surface. In the 

 following discussion, a theoretical derivation of the shadowing under simplifying assumptions 

 will be derived and compared with the model results. Let us remove the randomness and 

 assume that the keels are evenly distributed, 9.5 km, and have a depth of 4 m. These are the 

 means of the uniform and Rayleigh distributions from which the random sets were made. We 

 need to compute for plane wave angle 6 the fraction of surface not shadowed by the keels,/ 

 and convert to decibels by taking -10 log/ A genera) expression will be given first. Figure 10 

 shows the wave front, s, for a ray angle of 30 degrees for a specific keel. Rays that reflect in 

 front of and then strike the keel are represented by use of a reflection of the keel on the surface. 



Let d be the depth, b the half width of the keel, and 5, the length along the approaching 

 plane wave front that will intersect the keel. Then 



s t - [ (b 1 tan 2 8 * Ibd tan 6 * id 2 ) cos B]l 4d (2) 



The width of wave front that intersects the keel after reflection in front of the keel is given by 

 changing the sign of 6. 



Thus the total length of wave front intercepted. s t . is 



j, = [ (b 2 tan 2 * ^Jcos ) 2d (3) 



The surface length shadowed by this length of wave front is s t sin 6. These equations 

 apply up to the maximum slope angle of the keel, <*max- or 



« max = arctan(2</ b) (4) 



At greater values of 8. the shadow is just the width of the k ■■*!. 



10 



