## Resonance

Objects have frequencies that they prefer to vibrate at. If you
try vibrate it at a different frequency, the vibrations will be
dampened and eventually die out. If you try to vibrate it at its
preferred frequency, the vibrations will be reinforced and the
object will **resonate**.
Some examples of resonance:

- a standing wave on a skipping rope.
- the note you get when you blow into a half-full beer bottle
- the vibrations in the sounding board of a violin
- a swing swinging higher when you push it just right,
or you "pump" just right while you're sitting in it

### Resonance in a half-open tube

A tube that vibrates at one end and is open at the other (e.g., a
clarinet, the vocal tract) also has preferred frequencies.
You can get a standing wave in a half-open tube if the area of
high-pressure reaches the open end at exactly the same time the
closed end returns to normal pressure.

When this happens, the "reflected" waves travelling back from the
open end will exactly coincide with the waves travelling forward
from the closed end and they will reinforce each other. The tube
will resonate. (At a non-preferred frequency the backward-moving
waves will sometimes reinforce, sometimes cancel out, the
forward-moving waves, and you won't get a standing wave.)

The preferred frequencies for a half-open tube will
be all those frequencies (call them X) such that:
the length of the tube is 1/4 the wavelength of X, or the length of the tube
is 3/4 the wavelength of X, or the length of the tube is 5/4 the wavelength
of X, and so on. (This is often called the "odd-quarters law".)
This means the second resonating frequency will be three times higher
than the first, the next will be five times higher, and so on.

For a half-open tube that is 17 cm long (a typical length for an
adult male's vocal tract), the preferred frequencies are 500 Hz,
1500 Hz, 2500 Hz, 3500 Hz, and so on.

We often diagram the **frequency response curve** of a tube.
This shows for each frequency how much a tube would resonate
*if* you gave it vibrations at that frequency. The
frequency response curve for a 17 cm long vocal tract held in
neutral position (i.e., the position for schwa) looks like:

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