# The Physics of Closed End Tubes

### Essay Number Two on The Physics of Wind Columns by N. Drozdoff

Nicholas Drozdoff ndrozdoff@hotmail.com

### Mr. D's Music

The Physics of Closed End Tubes
A Continuation of the Physics of Open Ended Tubes

Let me begin by saying that if you haven't read the article "The Physics of Open Ended Tubes" you should. It will give you the basics and I am going to build off of that article here. This is the reason that this essay is a bit shorter.

All of the physics of a closed end tube behave exactly the same way in an open end tube with the exception of the reflection of the wave as it bounces off of the closed end. That merits some discussion here.

At the closed end the pressure must remain that same for the reflected wave as it was for the incoming wave. Let me explain. No energy or sound can get through the barrier (ideally), so there is no acoustic leakage here. The walls of the tube and the wall that constitutes the closed end a capable of holding the air in place. So when a high pressure front moves in it can't move into the room because the closed end holds it there and it can remain a high pressure. Now the wall and the air molecules dutifully obey Newton's Third Law which states that "for every action there is an equal and opposite reaction." This means that as the incoming wave hits the closed end it applies a force to the closed end (action). In turn the closed end applies a force to the air molecules (reaction) in the incoming wave. This means that the wall pushes the air molecules back down the tube. Similarly, if a low pressure front hits the closed end, it pulls against the wall which, in turn, pulls back on the air.

Now, what this means, if you stop to ponder this quietly for a moment, is that the reflection from the closed end is uninverted, which is to say, that if a high pressure pulse hits it, a high pressure pulse reflects from it and if a low pressure pulse hits it a low pressure reflects from it.

Now, you'll recall from the article/essay on the Physics of Open Ended Tubes, that it is a system of inversions and reflections that determines how we must periodically repeat the input of a pulse into our cylindrical pipe in order to continually reinforce the wave, thus producing resonance.

Now, you'll again recall from the previous article that with a tube with open ends, we can cause resonance at various frequencies. Specifically we can get resonance at a fundamental or lowest frequency for which the pressure is always at the room pressure at the ends of the tube (pressure nodes at the ends of the tubes). We can do it again at twice the fundamental, thrice, four times, etc. In short, it resonates at all of the integer multiples of the fundamental note.

Now with the tube closed at one end, this changes things slightly. First, for the fundamental, we must wait for the pulse that we have input into the tube to make another round trip before we repeat our input to help it out. This is because the reflection at the closed end doesn't invert as it did with the open end. This means that the fundamental of a closed end tube is an octave lower than that of an open end tube.

Next, due to this odd symmetry, the next note or frequency for which this combination of inverted and uninverted reflections allows is three times the fundamental. If we carry out this way of thinking, we can see that a closed end cylindrical tube will generate an overtone series based on odd integer multiples of the fundamental frequency of the tube. This is, indeed, the signature of a closed end cylindrical resonator; odd multiples of the lowest note that fits.

As an aside, if we think about thins a bit, we can see that the lowest note that fits an open ended tube is such that the wavelength of the note is twice that of the tube itself (a half wavelength regimen) whereas the closed end resonator's lowest note has a wavelength four times the length of the tube (a quarter wavelength regimen).

Now,if we try to play a closed end tube like a piece of pvc pipe like a trumpet, buzzing our lips into it (it's tough, but it can be done) or cylindrical bugle will produce a very weird set of notes which aren't very musical. There is an Australian instrument that behaves this way. It is called a didjeridoo. I have made one out of a three foot length of 2 inch pvc cylindrical pipe. It works quite well. There are some web sites on this instrument. You'll find them on my interesting links page.

So, in conclusion, a closed end pipe resonates with a fundamental which is an octave lower for a similar length of open ended pipe. Also, it produces only odd integer multiples of the lowest note.

You might want to review the discussion of the Overtone Series at this point and then move on to the article "The Physics of Brasses."

##### QED

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