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:
- 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.