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Suppose I am playing a hypothetical woodwind instrument with both ends open. Somewhere along the length of the tube is a hole, which cannot be at or near the exact center of the tube for the sake of this question. If I were to blow into the instrument with this hole open, the frequency of the standing wave between the hole I blow into and the hole along the tube would measure less than the standing wave between the hole along the tube and the open hole at the end of the instrument. If the hole was in the center, the waves would be the same length, as it's just dividing the instrument into two. I shall provide a visual representation of what I mean: enter image description here

  • The black lines represent the tube
  • The dotted black lines represent the open ends of the tube
  • The blue waves represent the standing waves of the instrument
  • The red circle represents a possible location for the hole

Ignore the fainter, numbered black lines

So now that you hopefully understand what I mean, what frequency will resonate? Am I doing something wrong with the standing wave? I'm not asking for a specific answer, I'm asking for the abstract mathematics behind it, if you know what I mean.

Thank you!

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A standing wave is the result of a reflected wave interfering on itself, and for a tube the energy, depending on its size is concentrated at frequencies that are integer multiples of the tube length. The energy, no matter where it is introduced will find the path of least resistance. If you introduce energy into the tube resonator where you have drawn the hole, and create separate standing waves, they will interfere, but will not sustain a resonance since they are not of the same frequency. If however the hole is large enough to act as a boundary it may sustain a resonance with one of the two ends, but not both.

Studies using a Rijke tube have shown the occurrence of resonance does depend on where the thermal energy is introduced, but it's a nonlinear, and complex mapping.

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