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

160 8. Acoustic Analogs, Ducts, and Filtersmust occur at the two junctions of the pipe. This results in a power-transmissioncoefficient expressed as follows:4T p =(A14 cos 2 kL +A + A ) 2sin 2 kLA 1(8.27)In Figure 8.5, Curve 2 constitutes the plot of Equation (8.27) for the same filtersystem used to obtain Curve 1. At low values of frequencies, i.e., kL ≪ 1, thetwo curves basically coincide. Equation (8.27), which is physically more valid,exhibits a minimum transmission coefficient of( ) 2A1 A 2T p (minimum) =S1 2 + (kL = π/2) (8.28)S2for the case where the length of the filter segment equals a quarter wavelength.Beyond this saddle point, T p gradually rises with increasing frequency until itreaches 1.0 (100%) at kL = π. At even higher frequencies the transmission coefficientvacillates through a series of maxima and minima until ka (a is the radiusof the through pipe) becomes somewhat larger than unity. From this point on,the transmission coefficient remains at unity. This trait of the transmission coefficientreaching a plateau of unity is also shared by high-pass and band-passfilters.Equation (8.28) may also be used to treat the constriction-type low-pass filterillustrated in Figure 8.6, since it does not matter if A 1 is larger or smaller thanA. The decrease in the area can be viewed as introducing an inertance in serieswith the pipe, but the validity of this analog also extends over a limited range offrequencies, as with the case of the expanded-area low-pass filter of Figure 8.5.In the real world of filter design (of mufflers, sound-absorption plenum chambersfor ventilating systems, etc.) the filter cross section cannot be radically differentin value from the cross-section area of the pipe. As is demonstrated in the curvesof Figure 8.5, a limited range of frequencies exists for the practical operation ofthe filters.Figure 8.6. A pipe with a constriction and its electrical analog.