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 λ in the above case. If the length of the tube is l, λ is 2l because a half wave is formed in the tube, that is,
$$\lambda = 2l.$$
Given the speed of sound, c, the frequency of sound, f, becomes
$$f = c / \lambda,$$
and therefore,
$$f = c / (2l)$$
For example, when c = 340 m/s and l = 17 cm,
$$f = c /\lambda = 340 / (2×0.17) = 1000 \mathrm{Hz},$$
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 λ matches the length of the tube, l, that is,
$$\lambda = l,$$and therefore, the frequency, f, becomes
$$f = c / l.$$
Likewise,
in the above case,
$$\lambda = 2l / 3$$and the frequency f becomes
$$f = 3c / ( 2l ).$$Finally, the resonance frequency, $f_{n}$, becomes
$$f_{n} = c / ( 2l ) × n \ (n = 1, 2, 3,\cdots).$$
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).
- 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.