Harmonics: Putting the standing wave in normal mode

When the oscillation of a diaphragm coincids with the antinode (greatest amplitude) of a satanding wave, the standing wave is in a normal mode. Normal Modes occur wherever there are standing waves, such as the ones on a vibrating string or in the pipes of an organ.

Suppose you have the closed-tube setup “The setup: Getting identical waves going in opposite directions.” You set a speaker to make a pure tone with wavelenght that makes a standing wave in the tube, which has an antinode at the diaphragm . The iscukkation of the diaphragm with the antinode of the standing wave, so it’s in a normal mode.

Standing waves are in a normal mode when the speaker makes a soind with wavelenght , which is given by

     (n=1,3,5,…)

Where L is the lenght of the tube and n is a whole number that labels the various normal modes.

The frequencies of normal modes of vibratuib are called harmonics.  The first harmonic (n=1) is called the fundamental frequency. Musicians often call the frequencies the higher frequency modes overtones.

Note that n must be an odd number because there are an odd number of a qyarter wavelenghts from the barrier (the closed tube) to an antinode. As n increases, so does the number of nodes and antinodes in your normal mode. So when your speaker makes a sound with 4L, there’s only one antinode, which is the one at the diaphragm.

The Following figure shows some normal modes of your tube- the two lines show the two positions of maximum displacement of the normal mode. Here, the horizontal axis measures distance from the diaphragm, and you can see the position of the closed end of the tube on the right side of the graph.  

Now try a concrete example- what are the first notes your tube likes to play? Suppose your tube is 0.983 meters long. The wavelenght of its normal modes are

If the speed of sound in your tube is 343 meters per second, then the frequencies of these modes are,

This means that the frequency of the lowest normal mode is 87.2 Hertz.

The next normal mode, when n=3, has a frequency of 262 hertz, which is about middle C on a piano. Where would you have to place your ear in the tube in order to hear silence when you play middle C on your speaker? You’d just have to listen at a node, which happens every half-wavelenght from enclosed end of the tube. When n=3, your wavelenght is given by

So you’d have to place your ear half this distance from the closed end of the tube- that is, 0.655 meters. For this normal mode, there are no other nodes in the tube (except, of course, the one at the closed end of the tube), so this is the only place you’d get silence.

It turnes out that any possible vibration of sound in the closed-tube-and-diaphragm setup is simply an interference of normal modes! So even the craziest, most complicated, erratic vibration can be boiled down to a matter of how much of each normal mode you have. This understanding comes from mechanics, particles (like electron) are allowed to be only in certain particular states. These states are like the normal modes of your tube. Like your tube the particles can be in a state that’s an interference of these normal modes. But when you actually measure the state of the electron, say, you can only ever see it in one of the normal modes! This is just a hint of some of the quantum weirdness that lies ahead for you in physics.