Stationary (standing) waves can occur in columns of air.
The frequencies at which resonance occurs depend on

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

• Resonance in Open Pipes

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