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When playing woodwind instruments, e.g., flute, if one blows harder, the sound will be one octave higher. Even harder gives even higher overtones. Does anyone know why?

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marked as duplicate by Ben Crowell, Kyle Kanos, Brandon Enright, John Rennie, Qmechanic Jul 30 at 7:45

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Closely related: Why does the fundamental mode of a recorder disappear when you blow harder?. Not sure if it a duplicate or not. –  dmckee Jul 30 at 0:39
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A crude overview: The flute's musical range extends up to three registers, starting from $B_3$. The way one switches between the registers is usually done in three ways:

  • The blowing pressure
  • The length of the air jet
  • The area of the lip opening

For example the most efficient way to increase the fundamental frequency is by extending the time for the air jet to travel across the embouchure hole until it is about half the period of an oscillation.

Think of the flute as a cylindrical pipe with both ends open. Thus its possible vibration modes form a series of frequencies that include all harmonics of the fundamental. Now by shifting the fundamental further to higher frequencies, higher registers can be reached. This can be accomplished by changing the blowing pressure, which in turn influences the resonance frequencies of the pipe. Roughly the frequency of the lowest mode is defined as (I may be wrong with the factor, one has to properly describe the possible standing waves to know the right factors): $$f_{low} = \frac{1}{T_{low}} = \frac{v}{4L} $$ Where $T$ is the period of vibration, $f$ its frequency, $L$ the length of the flute and $v$ is the speed of sound. The higher modes will be odd-numbered harmonics (of $f_{low}$) if I'm not mistaken. So you see that by increasing the blowing pressure, the air's jet speed is increased, so it's as though the you reduce L, or more rigorously you reduce the time of travel of jet waves from one end to another, this way the periods are shortened and higher frequencies can be matched.

All musical instruments have a lot of intrinsic details, details that at the end allow for an easier control over the created sounds. But for our purposes here, to summarize, woodwind instruments use feedback from an oscillating air column to control the flow of air into the instrument, and for the flute it's all about the air jet, air speed and the size/spacing of the holes to determine the cutoff frequency. Further resources: Flute acoustics, Wikipedia, Physical aspects of flute playing

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But if your explanation is correct, v should gradually increase when you gradually increase the blowing pressure. However, my experience is that when you blow harder and harder, the tone will first remain the same until a certain critical pressure above which it will suddenly jump to the first overtone. Why? –  user139981 Jul 30 at 1:18
    
No, be careful, $v$ in the formula is the speed of sound, which I assumed constant in this scenario. In the text I precisely mentioned speed of air in the jet. How fast the air goes back and forth in the pipe... –  Phonon Jul 30 at 1:39

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