The Theory of the Herschel-Quincke Tube

The well known simple explanation of the Herschel-Quincke interference tube is inadequate. It is herein shown that the ratio of the transmitted to incident acoustic energy is $\left[4\right(\frac{sin({\alpha }_3+{\alpha }_2)}{2}\left)\right(\frac{cos({\alpha }_3-{\alpha }_2)}{2}\left)\right]{\{{[1-2\mathrm{}cos({\alpha }_3+{\alpha }_2)+cos({\alpha }_3-{\alpha }_2\left)\right]}^2+4\mathrm{}{sin}^2({\alpha }_3+{\alpha }_2\left)\right\}}^{-\frac{1}{2}}$ wherein ${\alpha }_2$ and ${\alpha }_3$ are the phase changes over the two branches of the tube. This ratio is zero not only when ${\alpha }_3-{\alpha }_2=(2n+1)\mathrm{}\pi$, as formerly explained, but also when ${\alpha }_3+{\alpha }_2=2n\pi$, provided ${\alpha }_3-{\alpha }_2\ne 2n_1\pi$, where $n$ and $n_1$ are independent integers.