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

164 8. Acoustic Analogs, Ducts, and FiltersFigure 8.9. A ladder-type network used as a filter.all acoustic energy that transmits into the resonator returns to the main pipe in such amanner as to be reflected back from the source. The plot of the power-transmissioncoefficient in Figure 8.8 is fairly typical for a bandpass resonator.Equation (8.32) serves well for a resonator that has a relatively large neck radius.Narrower and longer constrictions will cause the transmission coefficient to deviatefrom the prediction of Equation (8.32), unless consideration is taken of the viscousdissipation that manifest itself more with such geometries.4. Filter Networks. The design procedure for acoustic networks, which incorporatesresonators, orifices, and divergence and convergence of pipe areas, is renderedeasier by analogy with the deployment of electronic filters. The sharpness of cutoffof an electrical filter system, for example, can be enhanced by using the laddertypenetwork of Figure 8.9. This network is constructed by using the reactancesof one type of impedance Z 1 in series with the line and the reactance of anothertype of impedance Z 2 shunted across the line. The standard theory of wave filtersstates that a nondissipative repeating structure such as that illustrated in Figure 8.9causes a marked attenuation of all frequencies except those for which the ratioZ 1 /Z 2 meets the condition0 > Z 1 /Z 2 > −4 (8.33)Several examples of acoustic ladder-type filters are displayed with the electricalanalogs in Figure 8.10. The condition of Equation (8.33) provides the followingcutoff frequencies:1f =4π √ MCfor the high-pass filter as shown in Figure 8.10(a), and1f =π √ MCfor the low-pass filter as shown in Figure 8.10(b). The behavior projected by thefilters as shown in Figure 8.10 applies only to wavelengths that are large comparedto the dimensions of the filter. At higher frequencies, deviation of the behaviorpredicted by electrical analogs becomes more prominent, because the filter begins