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

158 8. Acoustic Analogs, Ducts, and FiltersSolutionIn this case we consider this pipe to have two branches, one corresponding tothe given impedance and the other to an infinite pipe which does not provide anyreflections but does present an impedance of ρ 0 c/A. Applying Equation (8.20) weobtainba =ρ 0 c2Aρ 0 c2A + Z g(8.21)The ratio of pressure amplitude a t of the wave transmitted beyond x = 0 along theinfinite pipe to the pressure amplitude of the incident wave is found by insertingEquation (8.21) into Equation (8.18), yieldinga ta =Z gZ g + ρ 0c2A(8.22)We resolve the acoustic impedance Z g of the branch into real and imaginary components,i.e., Z g = R g + iX g . The reflection and transmission coefficients become( ρ0 c) 2R p =b22A∣ a ∣ ( ρ0 c) 2(8.23)2A + R g + X2 g T p = ∣ a t∣ 2 Rg 2 =+ X g2(a ρ0 c) 2(8.24)2A + R g + X2 gThe portion T pg of the power that is transmitted into the branch is T pg = 1 − R p −T p ,orR g ρ 0 c/AT pg = ( ρ0 c) 2(8.25)2A + R g + X2 gIf R g has a positive finite value, some acoustic energy is dissipated at the branchand some is transmitted beyond the junction, no matter what the value of X g is.When either R g or X g greatly exceeds ρ 0 c/A, all of the incident power is transmittedpast the branch. At the other extreme, when R g X g =∞, which correspondsto no branch, the power transmission is unity.8.5 Acoustic FiltersAdvantage can be taken of the fact that a side branch can attenuate sound energytransmitted in a pipe. The input impedance of the side branch determines whetherthe system can behave as a low-pass, high-pass, or band-pass filter. We shall nowconsider each of these filters.