Additional material - Resonance condition for pipes open at both ends
The wave from the source
Frequency of the sound source
A pipe open at both ends
More reflections at both ends of the pipe.
The wave is reflected at open end 2.
The wave is reflected at open end 1.
When the right-going waves are not in phase,
their interference results in a resultant right-going wave
with a small amplitude. Similarly, the resultant wave from the interference of
the left-going waves has a small amplitude. In this case, we do not have resonance.
At resonance, the right-going waves are in phase
and they interfere constructively
to produce a resultant right-going wave with a very large amplitude. This is also the case for
the left-going waves.
The resultant standing wave is the result of summing all right-going and
Similarly, we may add all the left-going waves to form a resultant wave moving to the
The resultant left-going wave
The resultant right-going wave
We may add all the right-going waves to form a resultant wave moving to the
At resonance, there is always antinode at the open end of the pipe since all the
right-going waves and the left-going waves have the same displacement at the
When a traveling wave in the pipe reaches an open end, part of it is reflected
back into the pipe and the other part is transmitted. The reflected wave
is not inverted upon reflection.
Let us follow a sinusoidal traveling sound wave as it travels in the pipe.