|Physics of the Didgeridoo - End Correction|
This page is written by Professor Ulf Sandberg to expand on our Physics of the Didgeridoo page. Ulf has correctly pointed out that our formula is rather simplified. We did spell out that it is for PVC pipes only, but it is great that some people have risen to the challenge to include conical shaped didjes. Professor Sandberg also tackles the problem of the "end correction" in below paper. "End correction" is dealing with the fact that the resonating air column does not stop at the end of the didj, but continues for a bit further.
Professor Ulf Sandberg:
My interest in this issue is because I am a researcher working with tyre/road noise reduction. Many tyres have in their tread patterns longitudinal grooves that more or less effectively form a pipe in the tyre/road interface, which causes noise emission at its resonance frequency, usually around 800-1200 Hz. The theoretical frequency of this phenomenon can be calculated much the same as the formula you have on your web site. However, the "tyre pipe" is open at both ends which gives different frequencies than if it were closed at one end like a didj. But even in tyres, there are "pipes" with closed ends, for example grooves that go from near the centre of the tyre tread out to the sides of the treads. Often, these are closed at the inner end. Thus, the closed pipe principle also applies to tyres.
Whether one pipe end is closed or not, the effective length of the pipe that determines the frequency is not exactly equal to the geometrical pipe length but slightly greater. The reason is that the air column in resonance protrudes a little outside of the geometrical end of the pipe. A few centimetres of the air outside the end takes part in the resonance behaviour, and this depends on the diameter. The larger the diameter, the more the resonating air column protrudes out of the pipe. Already Lord Rayleigh in his paper "On the Pitch of Organ Pipes" published in 1882 suggested a correction to the organ pipe length that depends on the pipe diameter. I call this "the end correction".
So how large is this end correction for diameter? Well, the actual correction is debated among scientists. Assume that the length of the pipe (the didj in this case) is L. Then the correction term is xd, where x is an empirically determined constant and d is the diameter (same unit is the length L). The effective length L' that shall be used in the formula that you published is then:
L' = L + xd
For a pipe closed at one end, the constant x is generally considered to be 0.3 - 0.4, most commonly 0.3. Assume you have a didj which is 100 cm long and a mouth (diameter) which is 4 cm. This means that xd will become 1.2 cm. This will increase the resonance wavelength and decrease the resonance frequency by 1.2 % in relation to using the uncorrected length L. If your didj were instead 8 cm (pipe diameter), the correction would be 2.4 cm which for a 100 cm long didj would be 2.4 %. The difference between the 4 and 8 cm didjes would then be 1.2 % in frequency. This is rather small but not negligible and may account for the observations you mentioned that you have made (but they could also be due to the flare correction that I describe hereunder).
However, in the scientific world (and even worse in the real world) things are not that simple. Your didjes do not have pipes with a diameter which is constant all over their lengths. They mostly have increasing diameters at the mouth; i.e., they are conical and display some kind of flare shape. Your didj ad079 is one of the most extreme examples of that. So which diameter shall one then consider? Well, Prof Neville Fletcher, working at CSIRO before his retirement, has studied the conical influence in some detail and he wrote a paper that includes this topic, "Acoustics of the Australian didjeridu", published in the Journal of the Australian Institute of Aboriginal Studies 1 (1983). Prof Fletcher who is one of the most competent acousticians that Australia has ever had has by the way written a 760-page book on musical acoustics. A more recent version of his 1983 paper, with the title "THE DIDJERIDU (DIDGERIDOO)", was published in Acoustics Australia, Vol 24, pp 11-15 (1996), and this one can fortunately be studied on the web. Look at http://www.phys.unsw.edu.au/~jw/dij/dij.html
Prof Fletcher describes the didj pipe in a more complicated but also more accurate way. Taking the changing diameter into account (see his paper on the web), the flare influence was shown by Prof Fletcher both theoretically and verified by experiments on didjes to give up to 25 % of increased frequency for a didj with a large flare. I call this "the flare correction". The "flare correction" results in a HIGHER frequency due to increasing open end diameter, while the above mentioned "end correction" has the contrary effect. For some didj shapes these two effects may in fact cancel each other although I think that the flare correction is generally the larger of them. In any case, the complicated but rather accurate formula for the resonance frequency that I recommend appears in the paper by Fletcher.
Prof Ulf Sandberg, Sweden
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