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

8.5 Acoustic Filters 163Figure 8.8. The effect of a Helmholtz resonator branch, resulting in band-pass transmission.both inertance and compliance, so it will behave as a band-pass filter. Apart fromalmost negligible viscosity losses, no net dissipation of acoustic energy occursfrom the pipe into the resonator. All energy absorbed by the resonator during somephase of the acoustic cycle is returned to pipe during other phases of the cycle soR g = 0. Denoting the opening area by A g = πa 2 , the neck length by L and thevolume of the resonator by V , the branch reactance X g is expressed asX g = ρ 0( ωL′A g− c2ωVwhich is then inserted into Equation (8.24) to yield the following transmissioncoefficient:⎡⎤−1T p = ⎢⎣ 1 + c 2( ωL′) ⎥(8.32)4A 2 − c2 ⎦A g ωVThe resonant frequency occurs when the transmission coefficient becomes zero,i.e.,√Agω 0 = cL ′ Vwhich corresponds to the resonant frequency of a Helmholtz resonator. When thisfrequency occurs, large volume velocities prevail in the neck of the resonator, and)