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I frequently use the tuned percussion tubes Boomwhackers in my elementary music teaching. I noticed something odd about them.

When I compare the smaller and larger C tubes, they sound one octave apart. The sound decays too quickly for me to verify the intonation with a tuner but it sounds like an octave to me.

I know that the wavelength of the two resulting sound waves should have a 2:1 ratio, as should their physical length. And it is nearly so... the large is 62.7cm and the smaller 30.3cm.

But why isn't it exactly 2:1? The small ones seem just a little too short. This must be on purpose -- the intonation is reasonably good. Also I have multiple sets to compare and so I know it's not a manufacturing defect.

These are the tubes. I'm using them with both ends open, if that matters.

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Elementary acoustical theory is based on the 'ideal string' with zero mass and no stiffness and the 'ideal tube' with zero diameter.

Real strings and tubes behave slightly differently.

The discrepancy in Boomwhacker lengths is doubtless something to do with End Correction, "...a short distance applied or added to the actual length of a resonance pipe, in order to calculate the precise resonant frequency of the pipe. The pitch of a real tube is lower than the pitch predicted by the simple theory."

https://en.wikipedia.org/wiki/End_correction

There is an article online specifically about the engineering of Boomwhackers, but unfortunately it's behind a paywall.

https://aapt.scitation.org/doi/full/10.1119/1.4862106

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    I wonder how diameter plays a role. Off the top of my head, I think most of the tubes are the same diameter; for some pitches, that width might correlate to a resonant frequency. Maybe for others the length has to be tweaked to compensate... Sep 4, 2022 at 14:18
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    The Wikipedia article includes a different, dead link to the same article, which can then be found in the Internet Archive Wayback Machine: "Boomwhackers and End-Pipe Corrections" by Michael J. Ruiz [PDF]: web.archive.org/web/20160826073345if_/http://www.mjtruiz.com/…
    – Bavi_H
    Sep 5, 2022 at 0:09
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    A good resource on end correction with insightful illustrations: pressbooks.pub/sound/chapter/vibrating-air-columns/… Sep 5, 2022 at 12:39
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    Using the formula on the Wikipedia page and demanding that the "effective" length of the lower tube is double that of the lower tube [2*(30.3 + 0.6 D) = 62.7 + 0.6 D], we conclude that the inner diameter of the tubes is about D ≈ 3.5 cm. This seems a little low to me (I'd expect more like 5–6 cm) but it's a plausible order of magnitude, at least. The Levine & Schwinger formula mentioned there gives a radius of about 3.5 cm which seems more plausible. Sep 5, 2022 at 13:02
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    @MichaelSeifert: About 1.75in (4.4cm), apparently - so not far off!
    – psmears
    Sep 5, 2022 at 13:21

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