The author's theory of acoustic transmission in a conduit with a branch line is extended by obtaining the thoretical values of the components of the point impedance of a closed and of an open tube.

The branch an open tube. The components of the impedance of the branch are $Z_1=\left(\frac{\rho \omega k}{2\pi }\right)D^{-1\mathrm{}}$ and $Z_2=\frac{\rho \omega }{c_0}+[\frac{\rho \omega }{k\sigma }·sin\mathrm{kl}coskl+\frac{\rho \omega }{c}({cos}^{2}kl-{sin}^{2}kl)-k\sigma \rho \omega (\frac{k^2}{4{\pi }^2}+\frac{1}{c^2}\left)sin \mathrm{kl} cos kl\right]D^{-1\mathrm{}}$, wherein $D=k^2{\sigma }^2(\frac{k^2}{4{\pi }^2}+\frac{1}{c^2}) {sin}^{2}kl+{cos}^{2}kl-\frac{2k\sigma }{c} sin \mathrm{kl} cos \mathrm{kl}$, $\rho$ is the density, $\omega =2\mathrm{}\pi \times \mathrm{frequency}=k\times \left(\mathrm{sound}\mathrm{velocity}\right)$, $c_0$ and $c$ are the conductivities of the conduit orifice and distant orifice respectively, $\sigma$ is the area and $l$ is the length of the tube. These values of impedances are used in computing the transmission through the conduit in accordance with fundamental theory (Phys. Rev. 5, 26, 688 (1925)). A comparison with experiment shows agreement. The transmission when the tube is at resonance is very different from the case of the same orifice without the tube.

The branch a closed tube. The above values of impedance reduce to $Z_1=0\mathrm{}$, and $Z_2=\frac{\rho \omega }{c_0}-\frac{\rho a}{\left(\sigma tankl\right)}$. The theory of transmission thus established is that of the Quincke tube. The selectivity is not sharp as is the cylinder in the open, the effect being marked over a total range of an octave. The effect of viscosity in increasing the minima of transmission is shown.

The branch an impedance of any kind. If the point impedance of the attachment be known, then to its imaginary component may be added $\frac{\rho \omega }{c_0}$ and both components used in the original formula for transmission. Also if such an attachment is by means of a tube of constant cross section, the theory will give the transmission.

• Received 5 January 1926

DOI:https://doi.org/10.1103/PhysRev.27.494