Influence of a Branch Line Upon Acoustic Transmission of a Conduit

General theory of transmission.—The ratio of the energy of the incident acoustic energy transmitted with the branch present to that with the branch absent is found to be $\left[\right(Z_1^{}{}_{}{}^2+\frac{Z_{1\rho a}}{2S}+Z_2^{}{}_{}{}^2)+{\left(\frac{\rho a{Z}_2}{2S}\right)}^2]\times {[{(Z_1+\frac{\rho a}{2S})}^2+Z_2^{}{}_{}{}^2]}^{-2\mathrm{}}$ where the point impedance of the branch is $Z_1+i{Z}_2\rho$ is the density of the fluid, $a$ the velocity of sound and $S$ the area of the conduit. Application to the Helmholtz resonator and the cylindrical resonator. The transmission ratios found are respectively, ${\{1+{\left[4\mathrm{}{S}^2{(\frac{k}{c}-\frac{1}{\mathrm{kv}})}^2\right]}^{-1\mathrm{}}\}}^{-1\mathrm{}} \mathrm{and}{\{1+{\left(4\mathrm{}{S}^2\right)}^{-1\mathrm{}}{[\frac{k}{c}-{\left(\sigma tankl\right)}^{-1\mathrm{}}]}^{-2\mathrm{}}\}}^{-1\mathrm{}}$ where $k$ is $\frac{2\pi }{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}-\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}$, $c$ is conductivity of the orifice, and $V$ and $\sigma$ are respectively the volume and area of the branch. The theory can be applied to ascertain the transmission, when the acoustic impedance of the side branch is known or it may be used to measure the acoustic impedance of the side branch.