THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

12.16 Acoustic Barriers 313In the case of rectangular barriers, three values of d i ,orn = 3, will usuallysuffice to describe the shadow zone of the barrier. Through Equation (11.20) wecan write for the mean-square pressure pd2 2 due to the direct field as follows:pd2 2 = QWρ 0c. (12.60)4πr 2with r constituting the direct length from the source to the receiver. CombiningEquations (12.56) and (12.60) yieldsp 2 b2 = WQDρ 0c4πr 2= WQ B4πr 2 ρ 0c. (12.61)where Q B ≡ Q D is the effective directivity of the source toward the direction ofthe shadow zone. Inserting Equations (12.61) and (12.55) into Equation (12.50)results in( Qp2 2 = Wρ B0c4πr + 4 )2 Rand the corresponding sound pressure iswhereL p2 = L w + 10 logQ B = Qn∑i=1( Q B4πr + 4 )2 R(12.62)λ3λ + 20d i. (12.63)In English units, where distances are expressed in feet instead of meters, Equation(12.62) becomes( Q BL p2 = L w + 10 log4πr + 4 )+ 10. (12.64)2 RFor a rectangular barrier n = 3 in Equation (12.63), and the required path differencesare given by Equation (12.59).From Equations (12.52) and (12.62) we obtain the barrier insertion loss IL:⎛Q⎜IL = 10 log4πr + 4 ⎞⎝2 R ⎟Q B4πr + 4 ⎠ . (12.65)2 RThe above equation applies to either the metric system or English system. It is interestingto note that if the barrier is located in an extremely reverberant environment,such as an echo chamber, the acoustic field reaches the receiver by rebounding unabatedfrom the surfaces of the room to the degree that the effectiveness of the barrierin blocking the direct field becomes insignificant. Because 4/R ≫ Q/(4πr 2 )and 4/R ≫ Q B /(4πr 2 ) for a reverberant room, Equation (12.65) will give a valueof IL = 0 dB.