Is there a way to find the velocity of air in a flute?

[orangeraindrops]

Homework Statement

I am working on a math investigation which involves the physics of the flute and I can't move on unless I find an equation for the change in required velocity of the air stream to produce different frequencies. I have a feeling I might have to abandon this idea and do something else, because I can't find an equation that will give me the required air velocity to produce different notes. I know that it changes because I have always been told to use a faster air stream for higher notes. I found a thesis online that might be helpful, but most of the math goes over my head since I'm still in high school.

Homework Equations

/ variables[/B]
-Bernoulli's Principle
-Stream function?
There are not many values that I can find since I don't have equipment to measure anything. I know the frequencies and wavelengths of the notes and the distance to the first open hole for each fingering (which changes the length of the tube). I may be able to measure pressure, but I'm not sure if my school's sensors are sensitive enough to pick it up.

The Attempt at a Solution

The only equation I have found so far is Bernoulli's Principle, but I'm not sure if I can use it because I don't have data for the pressure and I doubt I will be able to get it. The thesis I read mentioned "differentiating the stream function" to find "the sum of the acoustic velocity and the transverse velocity of the instability wave", but I read about the stream function and I don't really understand it. Also, I don't think the velocity of the instability wave is the same as the velocity of the air jet.

In summary, my question is: Is there an equation I can use to calculate the required air jet velocity to produce different notes (frequencies) in a tube (flute)?

 

[haruspex]

I know that it changes because I have always been told to use a faster air stream for higher notes.

Yes, but not for the reason you might think.
I suggest you avail yourself of a flute and try blowing at different speeds on the same low note (as determined by fingering), then repeat at the same speeds on a high note.

 

[orangeraindrops]

Yes, but not for the reason you might think.
I suggest you avail yourself of a flute and try blowing at different speeds on the same low note (as determined by fingering), then repeat at the same speeds on a high note.

When I blow with faster air on the same fingering, it produces the harmonics of the note (higher pitches). I'm trying to find an equation to describe the different air velocities needed to produce each note (I wasn't originally thinking about harmonics, but it's the same idea.) Is it not airstream velocity that is producing the changes? I think flute players tend to use the phrases "faster air", "blow harder", "blow with more pressure" interchangeably and I am confused about which factors are actually involved.

 

[Nidum]

http://newt.phys.unsw.edu.au/jw/fluteacoustics.html

 

[haruspex]

it produces the harmonics

Right. It does not change the funamental pitch. Instead, it adds harmonics, or overtones.
I've not read it through, but http://newt.phys.unsw.edu.au/jw/loudness.html looks relevant.

 

[orangeraindrops]

Right. It does not change the funamental pitch. Instead, it adds harmonics, or overtones.
I've not read it through, but http://newt.phys.unsw.edu.au/jw/loudness.html looks relevant.

So the fundamental pitch is still present, and a faster airstream makes the harmonics more audible. However, it is my understanding that the harmonics are still different frequencies than the fundamental pitch. I'm assuming the only way to measure the airspeed required to produce these higher frequencies is to use equipment. I was hoping there was some way to figure it out using other variables. I do have access to a pressure sensor.

Thank you for taking the time to help me.

 

[haruspex]

the harmonics are still different frequencies than the fundamental pitch

Indeed, they are multiples of it. I would expect that, in theory, all harmonics are always present, but as the power increases a greater fraction of the power is invested in the higher harmonics. But I've no idea what the formula for the distribution would be.
Was the link any use?

 

[Mister T]

When I blow with faster air on the same fingering, it produces the harmonics of the note (higher pitches).

I believe that's called overblowing. When a note is played you produce not just the fundamental, but the entire series of overtones, each with a different loudness. Normally the fundamental is the loudest, but when you overblow you increase the loudness of the first overtone, so that's the pitch you hear.

It's all a bit tricky, though, because if you leave the loudness of each overtone alone, and remove the fundamental, you still hear the pitch associated with the fundamental!

 

[orangeraindrops]

Indeed, they are multiples of it. I would expect that, in theory, all harmonics are always present, but as the power increases a greater fraction of the power is invested in the higher harmonics. But I've no idea what the formula for the distribution would be.
Was the link any use?

The link gave me a better understanding of how harmonics work, but I didn't find anything that answers my original question. I think I'll have to change my topic, which is really frustrating because this relationship was just supposed to be a small part of it and the rest of it worked well. I've looked at many websites and asked on a few other message boards, but there doesn't seem to be a straightforward solution.