Rijke tube: Why Sound is maximum when heated 1/4th length from bottom

[ayushmorx]

Why is it that the maximum sound coming from a Rijke tube happens only when heated wire gauze is placed at 1/4th the length of the tube from the bottom. According to Rayleighs criterion(copy pasted)

"If heat be periodically communicated to, and abstracted from, a mass of air vibrating in a cylinder bounded by a piston, the effect produced will depend upon the phase of the vibration at which the transfer of heat takes place. If heat be given to the air at the moment of greatest condensation, or be taken from it at the moment of greatest rarefaction, the vibration is encouraged. On the other hand, if heat be given at the moment of greatest rarefaction, or abstracted at the moment of greatest condensation, the vibration is discouraged"

Wouldn't the sound be maximum if heated gauze is placed at the bottom, where there exists an antinode?

 

[Philip Wood]

The Rijke's tube is open at both ends. The fundamental standing waves has displacement antinodes at each end and a single displacement antinode, in the middle of the tube. There will be pressure nodes at the end and a pressure antinode in the middle.

As the excellent Wiki article explains: "Most heat will be transferred to the air where the displacement of the wave is a maximum, i.e. at the end of the tube. However, the effect of increasing the pressure [which is what sustains the oscillation] is greatest where there is the greatest pressure variation, i.e. in the middle of the tube. Placing the gauze midway between these two positions (one quarter of the way in from the bottom end) is a simple way to come close to the optimal placement."

 

[Philip Wood]

ayushmorx: Did the explanation help?

 

[ayushmorx]

I really appreciate the response but what I still don't understand is this- that a rarefaction in a longitudinal wave is a region of high displacement(displacement antinode and pressure node). The Rayleigh's criterion mentions greatest rarefaction, which is obviously a displacement antinode. The position of the pressure node or antinode should have zero consequence on the problem(according to the statement).

Now the 2nd line i.e. "The effect of increasing the pressure...". The heat from the gauze, as I understand, expands the air near it which increases the displacement of the wave, and thus maximizes and sustains the oscillations. Why the heat intake of the wave maximum at maximum pressure variation, I just can't get at.

Basically I can't understand the role of the pressure gradient in the heat transfer from gauze to wave.

 

[Philip Wood]

a rarefaction in a longitudinal wave is a region of high displacement(displacement antinode and pressure node). The Rayleigh's criterion mentions greatest rarefaction, which is obviously a displacement antinode.

No. We are dealing with stationary sound waves. At a pressure node, the pressure stays the same all the time; there is no variation of pressure. But the variation of displacement is greatest here; a pressure node is a displacement antinode. At a pressure antinode the variation of pressure is greatest, that is from greatest compression to greatest rarefaction to greatest compression and so on. A pressure antinode is a displacement node (because when air one side of the pressure node is moving towards it, air the other side is also moving towards it, so air at the pressure node is not moving at all, and when air one side of the pressure node is moving away from it, so is air the other side of the pressure node so air at the pressure node isn't moving at all).

"The effect of increasing the pressure...". The heat from the gauze, as I understand, expands the air near it which increases the displacement of the wave, and thus maximizes and sustains the oscillations. Why the heat intake of the wave maximum at maximum pressure variation, I just can't get at.Basically I can't understand the role of the pressure gradient in the heat transfer from gauze to wave.

Understanding how the tube itself works is more difficult, and I don't claim to understand it properly. My understanding, such as it is, comes from the Wiki article. I think the gist of it is that transfer of heat from the gauze to the gas is greatest where the gas is moving most, that is at a displacement antinode. But the effect of the hot gauze on sustaining the oscillations would be greatest at a pressure node, so the best position for the gauze is a compromise between these two conflicting desiderata, that is about half way between a displacement antinode and a pressure antinode.

 

[ayushmorx]

Thanks a lot! I just did not notice anywhere that pressure node and antinode referenced pressure variations and not actual pressure values. That's why my simulations were giving me carnot-cycle defying values lol. Huge thanks!