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I have a simple recorder, like this:

enter image description here

When I cover all the holes and blow gently, it blows at about 550 Hz, but when I blow more forcefully, it jumps an octave and blows 1100 Hz.

What's the physical difference between blowing gently and blowing forcefully that the recorder suddenly jumps an octave and the fundamental mode is no longer audible?

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I doubt that the fundamental diappears. –  Georg Sep 15 '11 at 17:11
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Well, the following are true: 1) I cannot hear the fundamental. 2) A tuning device thinks the pitch jumps an octave. 3) I fourier-transformed the signal, and the 550 Hz signal is so small I can't even tell where it is, while the 1100 Hz signal is a huge spike. –  Mark Eichenlaub Sep 15 '11 at 17:26
    
I this case I have to revoke. I thought the fundamental might be just weaker than the octave. –  Georg Sep 15 '11 at 17:41
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It must be a non-linear effect. This paper describes some of these, but it seems, as far as I could gather from gleening over it, that the second harmonic is usually weaker. –  Raskolnikov Sep 15 '11 at 17:49
    
""It must be a non-linear effect."" Is there any method to create oscillations without nonlinear means? The paper You name is really excellent and explains everything, if You do not overlook that "" The method cannot be justified for a general case, since in principle each harmonic interacts back upon the jet to modify the source spectrum, but in the present case it represents a good approximation."" This feedback of course is essential for the sound generation. –  Georg Sep 15 '11 at 18:53
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4 Answers

up vote 5 down vote accepted

Overblowing is a phenomenon that exists in all wind instruments. The details of the physics are different from one instrument to the next, but there is a broad similarity, which is that it's the result of a nonlinear interaction between the air column and whatever is driving the air column.

The recorder is in fact one of the simpler examples to understand. The mechanism that drives the air column is called an edge tone. The mouthpiece of the recorder contains a knife edge. The stream of air encounters the knife edge, but doesn't split smoothly onto the two sides. Instead, it forms a vortex which carries the energy to one side of the edge. However, a feedback process then causes this pattern of flow to deflect until it flips to the other side of the edge. So this is a highly nonlinear system. It's binary. Air either flows to one side of the edge or the other, and if we label the two states 0 and 1, we get a pattern over time that looks like 0000111100001111... You could graph it as (approximately) a square wave.

When this edge-tone system is coupled to an air column, it's forced to accomodate its frequency to the resonant frequencies of the column. For example, if you imagine a pulse emitted from the edge, this pulse then travels down the tube, is partially reflected at the open end, returns, and slaps against the air in the edge-tone system, influencing its evolution. There is a tendency for the edge-tone system's vibrations to become locked in to one of the resonant frequencies of the column. In overblowing, the pattern switches from 0000111100001111... to 00110011... The square wave doubles its frequency from the fundamental frequency $f_0$ to the first harmonic $2f_0$.

The original square wave contained Fourier components $f_0$, $2f_0$, $3f_0$, ... The new one contains $2f_0$, $4f_0$, $6f_0$, ... As you observed on the oscilloscope, $f_0$ is absent from the overblown spectrum. The ear's sensation of pitch is based on the frequency of the fundamental, so we hear a jump in pitch.

Bamboo flutes and whistles also use edge tones, so exactly the same analysis applies. I think the original classic work on this was an analysis of organ-pipe acoustics in a German-language paper by Cremer and Ising. In general, the edge-tone system could be replaced by a reed, lip reed (as in brass instruments), or air reed (flute). There can be overblowing at the octave, or, in instruments such as the clarinet that have asymmetric boundary conditions, at an octave plus a fifth (i.e., a factor of 1.5 in frequency). On the saxophone, for example, a skilled player using a stiff reed can overblow to frequencies corresponding to several higher harmonics beyond the first.

References:

Cremer and Ising, "Die selbsterregten Schwingungen von Orgel," Acustica 19 (1967) 143.

Fletcher, "Sound production by organ flue pipes," J Acoust Soc Am 60 (1976) 926, http://www.ausgo.unsw.edu.au/music/people/publications/Fletcher1976.pdf

Backus, The acoustical foundations of music, Norton, 1969, pp. 184-186.

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Thanks for the work and the references! Haven't read in detail yet, but look forward to it. –  Mark Eichenlaub Apr 28 '13 at 16:39
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This is a guess, I'm no expert in the physics of musical instruments.

My understanding is that for a recorder, the mechanism of excitation is turbulence in the jet of blown air, striking an edge, which then preferentially excites resonant modes. The turbulence spectrum will peak with eddies of some scale $l_{max}$, which moving at wind velocity $v$ will have characteristic frequency $f =v/l_{max}$. Presumably, when you blow harder you are increasing $v$, thus shifting upward the spectrum of turbulence exciting your instrument, until it surpasses the lowest resonant mode. Then the amount of spectral energy available to excite that mode falls off and it seems to disappear.

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This seems correct--- the higher Reynolds number flow will have a bigger high frequency component, but there seems to be a feedback mechanism between the resonant mode and the sound-producing region, because the effect sounds discrete--- you don't see a smooth transition as an overtone gets louder. This is also true in overtone production in brass instruments. The feedback is hard to understand for me, because it requires interaction of the standing wave with the mouthpeice. Perhaps it is transmitted by the instrument body? Can you excite overtones in a coca-cola bottle blown over the top? –  Ron Maimon Nov 15 '11 at 0:03
    
I found some references that kinda support the answer: hyperphysics.phy-astr.gsu.edu/hbase/music/edge.html and hyperphysics.phy-astr.gsu.edu/hbase/music/edge2.html –  Kostya Nov 16 '11 at 12:29
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The operating regime of a flue pipe such as a recorder is governed by the Ising equation (reference at end): $$ I^2 = WST * 2 * P / \rho / F^2 / H^3 $$ or equivalently

$$I^2 = WST * v^2 / F^2 / H^3 $$ where

$WST$ is the thickness of the flue (windsheet)

$ P$ is the blowing pressure

$ F$ is the frequency

$H$ is the cutup (mouth height)

$v$ is the initial windsheet velocity.

When $ I$ is between 2 and 3, there is optimum performance. When I rises above 3, the instrument overblows.

L. Cremer and H. Ising: „Die selbsterregten Schwingungen von Orgelpfeifen“, Acustica (1967/1968) 19, page 143–153.

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Does this answer the question? If so, it could use some more explanation. –  dmckee Apr 27 '13 at 23:54
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Perhaps it's good to think about why the recorder normally chooses, amongst all possible modes, the fundamental. And without knowing the reason, I'd say by intuition that, like in "real physics", the lower frequencies need less energy at the same air throughput. But the airflow around the labium can in principle phase-lock to the harmonics just as well, and once this happens the fundamental will be eliminated. But why should it?
As long as the flow is locked to the fundamental and you're playing piano, the fundamental is dominant in terms of amplitude. There is always a certain amount of harmonics due to the nonlinearity of the nonlaminar flow, but at low volumes it is comparatively low. But if you now increase pressure, the nonlinearities will get ever stronger, and in general it will be the strongest frequency rather than the fundamental which is most likely to get phase-locked. Especially on cheap recorders, the fundamental may be damped be some resonance, so it can happen that the first harmonic ends up with more amplitude than the fundamental. And as I said, once this happens the overtone becomes stable and the new fundamental.

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It's not like I'm sure that this is all there is to the phenomenon, but I'm quite positive that it's a major contribution and it certainly can explain the effect. So, @Downvoter, would you care to point out what's wrong and, ideally, give a better explanation? –  leftaroundabout Sep 15 '11 at 23:02
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protected by Qmechanic Apr 27 '13 at 22:36

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