where c runs from - «» to + °> in D, i.e., above s » -q, above the branch 

 point 8 ■ -k and below the branch point s ■ +k . ThiB holds for y > 0, 

 while for y < \n use ♦ total ("y) " "^t tal^ y ^* The inte 8 ral here can 

 actually be evaluated in closed form, in terms of Fresnel integrals, 

 though the details are complicated, and are nothing to do with the 

 W-H method. We refer the reader to [5,10] for descriptions of the 

 details, and remark sirjply that the distant radiating acoustic field 

 can be estimated asymptotically from the formula 



£« 



s) exv(-isx - Y V) ds (8.29) 



/2k ir\ 



exp(ik r - TTiM) sin9 F(-k cos9) 



in which x ■ r cosS, y ■ r sin9, <. 6 <. n. (This formula may give 



apparent infinities in particular angular directions, and near those 



angles it is necessary to use more refined approximations.) To apply 



i; here we have to interpret (-k cos0 - k ) T , which we take as 

 i i oo 



-ik (1 + cos9) T because the arg of -k cos9 - k is equal to -it for 

 o o 



all 9 between and u. We also need 



, » i 



-i_ - (_ q - k )T - -i (q + k . )T § 



K_(-q) 

 again because arg (-q - k ) ■ -it. Then we have 



♦ -(A) 



2 1V o i ik r -Tfi/4 sin9/2 



(8.70) 



which gives the level and directivity of the scattered acoustic field. 



In the next section we look at the same problem, but with sub- 

 sonic mean flow over the surface. 



61 



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