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I have a complicated cylindrical geometry with a speaker mounted on somewhere middle of the rig. The one end is closed, the other is open. Sweeping the speaker frequency from 0 to 1000 Hz, I tried to find the acoustic resonance frequency of this geometry from several sensors at different location. It is found that there are some frequencies that give the local peak amplitudes. I thought those frequencies are the resonance frequencies, but when I compared the signals from two different sensors, the phase difference was neither in phase nor out of phase, but like 120 degree. This result is counter-intuitive to me because I thought the resonance always happens with the standing wave, which gives only in phase or out of phase between two sensors. If the end boundary conditions are hard or soft BCs, the resonance theoretically occurs with the standing wave pattern. So I think my open end is not actually sound soft BC, but rather have low impedance BC.
Anyway, the below is my questions:

  1. Does resonance always happen with the standing wave?
  2. What properties affect the end impedance. Is it function of temperature? Frequency?
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The idealized "open boundary condition" that is given in a first course on sound waves in pipes doesn't exist in the real world. The reason is that the pipe radiates sound into the environment, (otherwise you would not be able to hear any wind instrument playing a note!) and the sound wave outside the pipe is a traveling wave.

Note, the "end correction" that you have to apply to simple experiments on resonance in pipes is a way to model the effect of this sound radiation on the frequency of the resonance, but simple experiments don't attempt to measure the phase of the sound waves - they just assume there is "a resonance" and don't encourage the beginning students to ask too many questions about what is going on!

Therefore, the wave inside the pipe is some combination of a standing wave and a pure travelling wave, and the phase angle is not constant everywhere.

You can visualize the wave inside the pipe as a travelling wave moving towards the "open" end, but the reflected travelling wave in the opposite direction has a smaller amplitude, because some of the sound energy has been radiated away outside the pipe.

Of course the sound radiation out of the pipe also creates damping of the resonance response - which is why you need a continuous excitation source, otherwise the sound energy inside the pipe would decay to zero very quickly.

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