The tone of a tube closed at one end follows from f = v / (4L): the speed of sound divided by four times the length of the air column. With 343 metres per second and a tube of 0.40 metres that gives 214 hertz. Halve the length and the frequency doubles: exactly an octave higher.
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A tube of forty centimetres, closed at one end. Hit it and one particular tone comes out, always the same. How many vibrations per second is that? You can work it out, and you need only two things for it.
In three minutes you learn where that tone comes from and how you calculate it. What you hear comes not from the tube but from the air in it, and like everything else that can vibrate that air column has a frequency of its own. It depends on the speed of sound and on the length of the column, and those two sit together in one formula: the frequency is the speed of sound divided by four times the length. You see where that four comes from, namely that exactly a quarter of a wave fits in a tube closed at one end. Then you work the forty centimetre tube out to 214 hertz, with the two mistakes you can make there: forgetting the four, or turning the formula round. And you see what happens if you make the tube half as long.
For upper secondary physics, in the chapter on sound. This is the calculating part: why a tube resonates at all, and why two instruments on the same tone sound different, have separate videos on the channel.
In this video
The full explanation, in writing
The problem
0:00This tube is forty centimetres long and closed at one end. Hit it and you hear one particular tone. Always the same. How many vibrations per second are in it? Have a guess.
What happens in the tube
0:18What you hear is not from the tube but from the air in it. That air vibrates, and like everything else that can vibrate that air column has a frequency of its own at which it does so by itself. That frequency depends on only two things: how fast sound goes through air, and the length of that column.
0:39Longer columns give a lower tone, a shorter column a higher one. That is the same with every instrument you blow into: opening or closing holes changes the length of the air column, and with it the tone.
The formula
0:55Those two things sit in one formula. The frequency is the speed of sound divided by four times the length. `f` is the frequency in hertz, so the number of vibrations per second. `v` is the speed of sound in air, and at twenty degrees that is three hundred and forty-three metres per second.
1:14And `L` is the length of the vibrating air column, in metres. So in metres, and not in centimetres. That four seems to fall out of the sky, but it is there because in a tube that is closed at one end, exactly a quarter of a wave fits. Four of those quarters is a whole one.
Forty centimetres worked out
1:35Then the tube of forty centimetres. First to metres: forty centimetres is zero point forty metres. Four times zero point forty is one point sixty. And then three hundred and forty-three divided by one point sixty. That is two hundred and fourteen hertz.
1:52Two hundred and fourteen vibrations per second, and that is a tone you can simply sing back. If you get over eight hundred, you have forgotten the four and only divided by the length. And if you get something like zero point eight, you have done it the other way round: the length divided by the speed instead of the reverse.
What happens if you shorten it
2:15Look again at where `L` sits in the formula. It sits under the line. So the bigger `L`, the smaller `f`. Longer is lower, and that fits what you hear. Halve the column to zero point twenty metres, and the denominator becomes zero point eighty and four hundred and twenty-nine hertz comes out.
2:35Exactly twice as high. That is no accident. If you double the frequency of a musical tone, you get the same musical tone an octave higher. So a tube half as long sounds an octave higher. You can now work out for every tube closed at one end which tone comes out: speed of sound divided by four times the length, with the length in metres.
3:02Why that tube makes a sound at all and why it vibrates along at exactly that frequency, there is a separate video about that on the channel.
Frequently asked questions
How do you work out which tone a tube gives?
With f = v / (4L). Take the speed of sound, 343 metres per second at twenty degrees, and divide it by four times the length of the air column in metres. For a tube of 0.40 metres that is 343 divided by 1.60, which is 214 hertz.
Why is there a 4 in the formula f = v / 4L?
Because in a tube that is closed at one end exactly a quarter of a wave fits. Four of those quarters make a whole wave, so the wavelength is four times the length of the tube.
What happens to the tone if a tube is half as long?
The frequency doubles, so the tone goes exactly an octave higher. Halving 0.40 metres to 0.20 makes the denominator 0.80, and 343 divided by 0.80 is 429 hertz instead of 214.
Does the tube itself vibrate or the air in it?
The air in it. The air column has a frequency of its own at which it vibrates by itself, and that is what you hear. That frequency depends only on the speed of sound and on the length of the column.
What is the commonest mistake in this calculation?
Using centimetres instead of metres, which puts the answer a factor of a hundred out. After that comes forgetting the four, which gives over eight hundred hertz, and turning the formula round, which gives something like 0.8.
The length goes into the formula in metres, not in centimetres. That is the mistake that puts most answers a factor of a hundred beside the truth.
Physics with Thomas makes physics explainers for secondary school. No words on screen, so what you see works in any language.
About me
My name is Thomas Schuurmans, and I have a passion for physics and for understanding why things work the way they do. What I want is to get young people asking “why is that?” and “how does that work?” more often, and to hand them the tools of physics in a way that is simple and that you can see.
I graduated in applied physics at Delft University of Technology in 2003. After that I spent more than twenty years outside education: first at TNO, the Dutch applied research institute, then at a design agency, and eventually founding Proportion Global, through which I work on innovation questions in Africa, Latin America and South Asia. That work resembles physics more than you would expect: don't start from a solution, first understand what is going on, try something, be wrong, and look again.
Since 2026 I have been teaching physics to both lower and upper secondary classes, and I started a master's at the University of Amsterdam for my full teaching qualification. That is where I learned that a secondary school pupil's real attention for new material lasts about seven minutes. And my own weakness happens to be telling too many side stories.
That is where the idea came from: videos that explain one topic sharply and visually, inside those seven minutes. I make them for my own pupils. Then I publish them, because good explanation should be within reach of anyone who needs it, wherever you live and whatever language you think and speak in.