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Kundt's Tube: How to Investigate Sound Waves in Media

Important Formula

Formula: Kundt's Tube

$$L ~=~ \frac{ \class{blue}{\lambda} }{4} \, \left( 2 \class{red}{n} - 1 \right)$$ $$L ~=~ \frac{ \class{blue}{\lambda} }{4} \, \left( 2 \class{red}{n} - 1 \right)$$ $$\class{blue}{\lambda} ~=~ \frac{ 4L }{ \left( 2 \class{red}{n} - 1 \right) }$$ $$\class{red}{n} ~=~ \frac{ 2L }{ \class{blue}{\lambda} } + \frac{1}{2}$$

Rearrange formula

What do the formula symbols mean?

Tube length

$$ L $$ Unit $$ \mathrm{m} $$

The tube can only have certain lengths so that a standing wave is formed in the tube.

Wavelength

$$ \class{blue}{\lambda} $$ Unit $$ \mathrm{m} $$

Wavelength of the acoustic sound directed into the tube.

Number of nodes

$$ \class{red}{n} $$ Unit $$ - $$

A natural number $ \class{red}{n} = 0, 1, 2, \ldots $ that specifies the number of vibration nodes and defines the possible lengths of the tube.

The Kundt's tube is an experiment for investigating sound waves in various media, especially in gases. The basic idea is to generate a standing sound wave in a tube and to investigate the wavelength $ \lambda $ and frequency $ f $.

A typical Kundt's experiment consists of a long, thin tube filled with a fine powder. A sound is generated at one end of the tube, for example by a vibrating resonator or a loudspeaker. This sound creates a standing wave in the tube, in which sound waves are reflected by the medium in the tube.

Standing Waves in a Kundt's Tube

$$ \begin{align} L ~=~ \frac{ \class{blue}{\lambda} }{4} \, \left( 2 \class{red}{n} - 1 \right) \end{align} $$

$$ \begin{align} L ~=~ \frac{ c }{4 \class{green}{f}} \, \left( 2 \class{red}{n} - 1 \right) \end{align} $$

How does the pitch of a wind instrument depend on its length?

The longer a wind instrument is, the lower tones it produces! Take, for example, an organ pipe: It is, from a physical perspective, a one-ended open tube where standing sound waves can form, with the following wavelength $\lambda$:

$$ \begin{align} \lambda ~=~ \frac{4L}{n} \end{align} $$

Here, $ n ~\in~ {1,3,5...} $ is an odd natural number, and $L$ is the length of the organ pipe. The frequency $f$ (i.e., the pitch) of the sound wave and its wavelength $\lambda$ are linked through the phase velocity $c$:

$$ \begin{align} c ~=~ \lambda \, f \end{align} $$

Now, equation 3 can be substituted into equation 4, and rearranged for the frequency $ f $:

$$ \begin{align} f ~=~ \frac{n \, c}{4L} \end{align} $$

As seen in equation 5: The larger the length $ L $ of the wind instrument, the smaller the frequency (that is the lower the pitch).