There are many well-known homework assignments to find eigenfrequencies of an open organ pipe leading to the solution for the fundamental frequency of such a pipe:

$$f_0 = \frac{c_0}{\lambda_0} = \frac{c_0}{2L}$$

where $L$ is the pipe length.

But if you measure the real organ pipe or flute, the fundamental is lower, sometimes significantly (up to 20% deviation from theoretical value). Why?


That is because of the termination impedances at the pipe end and mouth. The above described relationship for fundamental frequency is given for zero termination impedances (ideally open pipe) which is not the real case.

The simplest way to account these impedances is to introduce corresponding length corrections, so for the $f_0$:

$$ f_0 = \frac{c_0}{2\tilde{L}} \ , \ \ \tilde{L}=L + \delta_e + \delta_m $$

where $\delta_e$ is the end correction and $\delta_m$ the mouth correction:

$$ \delta_e \approx 0.34\sqrt{S} $$

$$ \delta_m \approx 0.73\frac{S}{\sqrt{S_m}} $$

where $S$, $S_m$ is the cross sectional area of the pipe and the pipe mouth respectively.

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