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We already know that $v_s = f\lambda$ where $v_s$ is the velocity of sound wave. Now as per my teacher, if we use a resonance tube to calculate the speed of sound, we first measure the length say $l_1$ for the fundamental frequency and then we measure the length say $l_2$ for the 3rd harmonic frequency and since the resonance tube acts like a closed end organ pipe, $l_1 =\frac{\lambda}{4}$ and $l_2 = \frac{3\lambda}{4}$.

And then he used the equation :

$l_2-l_1 = \frac{\lambda}{2}$

$\implies l_2 - l_1 = \frac{v_s}{2f}$ and since in the experiment we know the frequency because of the tuning fork we used we can calculate velocity of sound by $2f(l_2-l_1)$. Now my question is that why not simply calculate velocity of sound using $v_s = f\cdot 4l_1$ or $v_s = f\cdot \frac{4l_2}{3}$ ? My teacher said that its because $l_1$ is actually effective length and so is $l_2$ and when we take their difference the end correction gets cancelled. But aren't we supposed to include end correction in the calculation ?

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  • $\begingroup$ After re-watching what he said, it appears that we actually don't know the value of the end correction so we can't use those two equations since we would need to include end correction (e) i.e. the correct equations would be the length we measure for $l_1$ + the end correction but we don't know it, so once we get the velocity of sound we can find it. Am I correct here ? or did I get it wrong ? $\endgroup$
    – Koustubh Jain
    Mar 13, 2021 at 5:34
  • $\begingroup$ Also one more thing since end correction (e) is $0.6r$ where $r$ is the radius of the tube why can't we include that as end correction ? $\endgroup$
    – Koustubh Jain
    Mar 13, 2021 at 5:42

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The end correction is sufficient for a number of applications, but it is actually an approximation that is strictly valid only for a certain idealized tube ending. If you are hoping to get a precise estimate of the sound speed, it would be nice to remove that uncertainty.

The end correction you specified, $0.6r$, is the end correction for an isolated and unflanged tube. If the tube is baffled, it goes to about $0.85r$. Furthermore, according to my textbook these end corrections are just upper-limit estimates. Depending on your tube, these uncertainties can pile up and make your sound speed estimate essentially useless.

As an aside, it is worth noting that the end correction is a low-frequency approximation for the radiation condition. Thus, if the wavelength becomes comparable or shorter than the radius of the tube the end correction will become frequency dependent, and even using your teachers method will not yield accurate results. Of course, if you have a wavelength on the order of the radius and you are still using planar duct acoustics for your analysis, you have other problems to deal with than just the end correction.

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