Acoustic Transmission with a Helmholtz Resonator or an Orifice as a Branch Line

The author's theory of acoustic transmission in a conduit with a branch line is extended to the cases of a Helmholtz resonator and an orifice attached as branches. The effect of viscosity is included in an approximate form.Helmholtz resonator. The components of the point impedance of the branch are found to be: Z₁=(2ωρμ)1/2(πR³)^-1L and ${Z}₂={{ρ}{ω}}{c}{-}{{ρ}{a}²}{V{ω}}+{(2{ω}{ρ}{μ})}1/2L{({π}{R}³)}^{{-}1}$, wherein ${ω}=2{π}×{}frequency$, ${ρ}$ is density, ${μ}$, viscosity, $R$, radius of orifice, $L$, its equivalent length, $a$, the velocity of sound, and $V$, the volume of the resonator. Experimental verification of the theory is obtained and in addition there is found the approximate value of a simple channel equivalent in viscosity effect to the orifice of the resonator. The serious diminution of transmission caused by the resonator is not limited to the neighborhood of the resonance frequency but extends over a range of two octaves.Orifice. The components of the point impedance of the branch are found to be the same as those of the Helmholtz resonator, but with the term containing V omitted. Experimental verification is obtained with sizes of orifices from 7 percent to 42 percent the diameter of the conduit. The radiation from such orifices is shown to be relatively small and thus their impedances are practically imaginary. Transmission in the conduit increases with frequency. Action of orifices in musical instruments is explained by the theory. Relative values of radiation and transmission are shown by computation.