Stationary (standing) waves can occur in columns of air.

The frequencies at which resonance occurs depend on

i) |
**
the length of the air column** |

ii) |
**
the speed of waves in the air column** |

Resonance in Closed Pipes

Graphical representation of an air column in a *closed*
pipe resonating at its fundamental frequency, f_{o}
(the *lowest* frequency).

Explanation of this
diagram

The distance between the full and broken
lines represents the amplitude of the oscillation of the air
at that point in the pipe.

*
**At the closed end*, waves are reflected**
****
with a phase change of 180°**,
there is *no displacement*: a displacement
**node**
exists at the closed end.

*
**At the open end*, the air is free to
move; waves are reflected with no
**phase change**
so a displacement
**anti-node**
exists at the open end.

Therefore, if waves take
**half a
time period**
to travel *twice the length* of the pipe,
resonance occurs and a loud sound is heard.

For the fundamental frequency, f_{o},
length of air column =
**
/4**

**
**
Therefore
= (approx)
**l/4
**and, as f = v/l
we have

The same pipe can be caused to resonant at
higher frequencies.

The diagram below represents the *second harmonic* in the
same closed pipe.

Now the waves travel twice the length of
the pipe in **1½ time
periods (3T/2)**.

Now,
= (approx) **3l/4**
and therefore

In general, for a closed pipe