III-113 



Let I denote the integral in (111-153). Since w (-9 ) = w (6 ), I can be written as 



1 n 



11 







/a e ^^ de(vdv) 



(III- 154) 



In evaluating the integral analytically, we assume y = in the expression for w. 

 This corresponds to a grazing angle of incidence for the incident sound . Letting 

 B^ = 1 - Y 2 , we then have w^ = 1 -I- B^ - 2B cos @ . With (B, 8 ) as the vari- 

 ables of integration, we have the domain illustrated below. 



Then B^ = 1 -I- w^ - 2w cos 2 and so v^ = 2w - w^ - 4w sin^ ^ . Writing 

 the integral I in terms of the variables w and , the element of area v dvd§ be- 

 comes w dw d (2 0). For any particular value of w, can range from to such 



that 1 = 1 -I- w^ - 2w cos 2@, i.e. 

 variables and limits, I becomes* 



2 0n 



to such that sin 0q=(2-R)/4. In terms of these 



I = 4 



/I 







7/3 



1 

 aw 



d dw 



(III- 155) 



*Marsh reports I 



2 ''O 



w 







-7/2 



V w 



d dw. Apparently a factor of 2 was 



omitted in reducing the range of the original variable 9 or in changing the variable 

 of integration . 



Arthur H.HittleJnt. 



S-7001-0307 



