General remarks

The laws of refraction and the concept of impedance matching can also be applied to acoustic propagation at the junction between two cylindrical pipes of different diameter.

When an acoustic oscillation propagates along the pipe, there is a pressure difference between neighbouring elements of the gas contained in the pipe and, correspondingly, a flow of the fluid molecules. Unlike what would happen for hydraulic flow in pipes, both these quantities oscillate, and so there is no macroscopic transport of the mass of the fluid along the duct, but only a temporary advance, followed by a return to the equilibrium position, a retreat, again a return to equilibrium, and so on. Nevertheless, the continuity conditions still hold along the duct. For example, the mass of air that passes from a small cross-section to a larger one through the junction point must remain constant. Since at the junction point the cross-section of the duct suddenly increases, for the mass to be conserved there must be a corresponding decrease in the flow velocity.

Let us try to clarify the process through some examples.

Examples

To fix ideas, we have built two examples in which a section of pipe 50 cm long with a radius of 2 mm is joined to another section also 50 cm long, but with variable radius.

If we gradually increase the radius of the second section of pipe, we pass gradually from a pipe 100 cm long with a radius of 2 mm to a situation in which a 50 cm pipe is joined to a larger pipe and, in the limit, when the ratio between the radii ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$ is very small (that is, the second section is much wider than the first), we simply have a 50 cm pipe that essentially opens into open space. As can be seen by examining the admittance graph, the resonances corresponding to the odd multiples of the fundamental frequency of the 100 cm pipe gradually disappear, and the response tends to become equal to that of an open-open pipe only 50 cm long. The fundamental therefore rises by an octave. The graph on the right, which shows the behaviour of the resonances, shows that the relative values of the admittance maxima change very little, although examining the amplitudes shows that the odd peaks are all strongly attenuated already for diameter ratios of 1/5.

Of course, when the radius of the second section becomes much larger than the wavelength passing through the pipe, our simple one-dimensional approximation breaks down, since the plane wave leaving the narrower section undergoes diffraction and turns into a spherical wave in the wider section. (On this subject see also the experiment on diffraction in the interactive laboratory)

Geometry	Acoustic response	Resonances
At the beginning the system corresponds to a single pipe 100 cm long with a radius of 2 mm. At the end it is physically equivalent to an open-open pipe 50 cm long with a radius of 2 mm.	The odd multiples of the fundamental frequency are strongly attenuated already for radius ratios of 1/5.	Note that the odd multiples of the fundamental frequency shift when the radius of the second section of pipe is very large. This shows that they are associated with this second section, while the even multiples of the fundamental frequency remain unchanged throughout the transformation.

In the second case we have a section 50 cm long with a radius of 2 cm joined to a section of the same length with decreasing radius. In this case too we start from the response of a single open-open pipe 100 cm long, and gradually move towards the case of a half-open pipe 50 cm long. The second tube in fact becomes more and more like an obstruction, and so tends to resemble a closed end. In this case the even resonances disappear, and, correctly, the odd multiples of the fundamental remain, while the fundamental itself stays unchanged.

Geometry	Acoustic response	Resonances
We gradually narrow the radius of the second section of tube until it becomes capillary, that is, almost indistinguishable from a real obstruction of the acoustic flow.	The odd multiples of the fundamental frequency of the whole pipe remain unchanged, while the even ones gradually disappear.	The maxima of the harmonics that are even multiples of the fundamental disappear for a diameter ratio of 7 to 1