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We have built a "hoot tube" or a Rijke Tube, which resonates at 70Hz (measured by microphone) due to convection currents. We want to know what is the forcing function of this system. Does the forcing function have to match the resonant frequency of the tube? If so, what is it that's oscillating 70 times per second?
Answer 1:

This is very interesting; I'd never heard of such a device before. I guess this is an area of active research as this thermoacoustic oscillation effect can cause problems in jet engines and such. I found some information on some interesting websites (search on Google for "Rijke Tube"):

Rijke Tube Resonator

As far as the forcing function goes, I don't think the frequency has to exactly match the resonant frequency of the tube. Perhaps you can fiddle with the element in the tube to change the frequency of the resulting oscillations upon heating to be closer to the resonant frequency. Then the sound might have a larger amplitude. I have to admit, though, that I don't completely understand how these things work. I'm not sure if the air passing by the hot element and expanding starts a cycle of pressure wave oscillations (a periodic forcing function) or if there is something more subtle going on.


Answer 2:

There are a couple of different ways of building a Rijke Tube and they all operate will similar principles. The most common construction is a vertical tube with a heat source in the bottom section of the tube.

The study of systems like this falls within the area called "Dynamical Systems and Control",which is rather mathematical. However I can give you some physical intuition about what is going on that should help with your explanation to your students.

One way of thinking of this is to compare your Rijke tube to a organ pipe. Anorgan pipe has a resonant frequency which depends on its length. When air is forced into it the pipe resonates and we hear the musical note. It's actually the air pressure that is resonating here. The pipe will contain regions of slightly higher and slightly lower pressure in an oscillating pattern. These pressure waves are broadcast from the pipe and travel through the atmosphere (at the speed of sound) to our ears. Our eardrums move in response to these pressure variations and we sense this movement as sound.

On the other hand the Rijke tube experiment is not stable in the same way. Without forcing air into it, the oscillations build up to some level and are sustained at that level. As in the case of a organ pipe, it is the pressure in the air column that is oscillating. A few other things are oscillating as well and I'll talk about these later.

It is the interaction between the heat source and the acoustic characteristics of the tube which creates this unstable response The size of the pressure variations (which we hear as the volume of sound) is ultimately limited by the available energy in the heat source and how well the heat source transfers this energy into heating up the air around it. The frequency of the oscillations (or musical tone) is determined by the length of the tube,just as it is in the organ pipe case.

We can look in more detail at the interaction between the heat source and the air pressure for a bit more insight. The heat source warms the air around it, and as the warm air rises in the tube it pushes up on colder air above it. The rising warm air also draws more air into to the tube past the heat source. This air flowing past the heat source cools it down very slightly, making slightly less heat available to warm this new batch of air. This batch of air is therefore cooler than the warmed air above it and the air flow into the tube slows. With the slower flow of air the heat source warms up again and the cycle begins again. This leaves an oscillating pattern of hot and cold air in the tube, and because hot and cold air have slightly different densities this means that we have an oscillating pattern of pressures in the tube. This pressure oscillation creates sound exactly like the organ pipe.

So there are several things that are resonating at 70 Hz in your tube. The obvious one is the air pressure in the tube, which leads to the sound that you hear. The preceding discussion suggests that the temperature of the heat source is also oscillating at 70Hz, as is the air flow into the tube. Resonant tubes (both Rijke tubes and organ pipes) are extremely efficient in terms of the size of the air pressure wave they create when they resonate. The heat source temperature variation is probably extremely small (perhaps a tiny fraction of a degree) which means that you would not be able to measure it easily. Similarly, the change in the air flow into the pipe is also extremely small. It would be very difficult to setup a school experiment to measure either of these quantities. You might consider a few modifications to your experiment to illustrate some of these points.

- You could try disrupting the air flow into the tube, perhaps by constricting it in some way. This will probably stop the oscillation if you are cut off enough of the air supply. If your heat source is a flame (rather than a small electric element) then you might end up also cutting off the oxygen supply and extinguishing the flame.

- You could also try using different heat sources, particularly ones with different thermal masses. To get an oscillation the temperature of the heat source must be able to change quickly (more precisely it must be able to change at 70 Hz). Something with a large thermal mass (for example an electric stove top heating element) would probably not be able to do this.

- Change the length of your tube; making it longer should make the note lower in frequency.



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