Resonance in a Uniform Tube - Part 1-

When blowing into one end of a long tube, it is possible to make a whistle sound. This is because a resonance occurs in the tube. Let's take a look at the resonance in a uniform tube whose cross-sectional area does not vary along the length of the tube. When a resonance occurs, a standing wave is formed in the tube. The velocity distribution of air particles in the tube is shown as follows:



When the both ends are closed, the both ends become the nodes. This is because air particles cannot move freely at the closed (fixed) ends.

Now, let's estimate the wavelength in the above case. If the length of the tube is l, is 2l because a half wave is formed in the tube, that is,
            = 2l.
Given the speed of sound, c, the frequency of sound, f, becomes
            f = c /,
and therefore,
            f = c / ( 2l ).
For example, when c = 340 m/s and l = 17 cm,
            f = c / = 340 / (2~0.17) = 1000 Hz,
and the resonance frequency is 1 kHz.

The following standing wave can also be formed in which a resonance occurs:



In this case, the wavelength matches the length of the tube, l, that is,
            = l,
and therefore, the frequency, f, becomes
            f = c / l.

Likewise,



in the above case,
            = 2l / 3
and the frequency f becomes
            f = 3c / ( 2l ).
Finally, the resonance frequency, fn, becomes
            fn = c / ( 2l ) ~ n (n = 1, 2, 3, ...).

Let's take a look at the resonance phenomenon in the closed uniform tube with Kundt's experiment. In this example, a tube with the length of 17 cm is used and a small amount of cork powder is inserted inside the tube. At one end, a loudspeaker is mounted and produces a sinusoidal wave (pure tone) originally generated from an oscillator. When the frequency of the sinusoidal wave changes, we can observe loud sounds and "dancing" cork powder at a certain frequencies. These frequencies correspond to the resonance frequencies of the tube. In the following video, the frequency was around 1 kHz).




[References]
Stevens, K. N., Acoustic Phonetics, MIT Press, Cambridge, MA, 1998.
Sakamoto, S., Asakura, T., Ueno, K., Sakimoto, Y., Satoh, F. and Tachibana, H., "Visualization of acoustic resonance phenomena using Kundt's dust figure method," J. Acoust. Soc. Am., 120(5), 3070, 2006.