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I have been doing a question based on a resonance column tube in which tunning fork of $800 \ \text{Hz}$ is used. The water level is changed, so as to attain different lengths which in turn also changes the mode of vibration. It asked to find the velocity of air in that setup which is rather doable. The problem arose when I put the same velocity in the expression $$\text{frequency}=\frac{(2n+1)v}{4l} $$ to find the mode of vibration but all I get are nonintegral values of $n$ where $n$ is the mode of vibration. I don't know what is going on.

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Once you have fixed $v$ then you have two options:

  1. Fix $l$ and vary $n$ to find the different resonant frequencies of a tube with length $l$.
  2. Fix the frequency and vary $n$ to find the different tube lengths that will resonate at that frequency.

If you set both frequency and $l$ to arbitrary values then you will almost always find that $n$ is not an integer which tells you that a tube with length $l$ does not resonate at that frequency.

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