Helmholtz resonance of a bottle and a closed tube approach

[coolparticle]

Hello everyone,
I have been doing some research about Helmholtz resonance of a bottle for a science competition, but I don't find it clear how the Helmholtz frequency in a bottle is connected with a concept of seeing a bottle as a closed tube of a length L and fundamental frequency f=v/4L ? I understand that the latter formula does not consider the shape of the bottle, while Helmholtz resonance does. When we pour liquid in the bottle and observe the change in resonant Helmholtz frequency with change in volume, we see that shape matters (I guess because if affects the amount of volume of air in the "resonator"). Still, if the bottle looked like an almost perfect cylinder and we poured water in it to observe change in frequency, would Helmholtz frequency (recorded and determined by computer) and f=v/4L (where L is the height of air column) give the same values?

Please, help me, I'm lost
Thanks!

Edited March 6, 2017 by coolparticle

[swansont]

Would (or can) the Helmholtz frequency reduce to the cylinder frequency for a cylinder? I don't see offhand how you can do that, since the resonator is relying on having a volume of air to compress. Where does the neck stop and the reservoir begin?

 

Also, the closed-tube resonance does not depend on the volume, only the length. It works because you are creating a standing wave with a node at the closed end. The part where the tube joins the larger volume is not treated as a closed end — quite the opposite. The resonator has a characteristic "springiness" of its air.

[studiot]

Just to make emphasize what swansont was saying.

 

A Helmholtz resonator works quite differently from a tuned pipe.

 

Essentially it works by bouncing (vibrating) the plug of air in the neck of the bottle up and down against the springiness of the air in the body of the bottle.

 

This is achieved by raising or lowering the pressure just outside the mouth of the bottle.

This causes the air in the neck to move in or out, pressurising or depressurising the air in the body of the bottle

The volume of air in the bottle opposes this and the resonance depends upon the volume of the bottle.

 

Note that it is not necessary for the bottle to be sealed or enclosed.

This is how a vehicle exhaust silencer works, by having changes in pipe size causing this effect on the exhaust airflow.

[Enthalpy]

The Helmholtz resonator is modelled as separated inductance (the neck) and capacitance (the belly), making it simple.

 

A quarter-wave tube has no distinct neck and belly. It is modelled accurately with inductance and capacitance elements distributed all over the length, with a computation a bit more complicated.

 

Though, there are similarities, because the the mouth of a tube, the pressure swing is small, so the capacitance has a smaller effect, and at the closed end, the speed is small, so the inductance has a smaller effect.

 

So you get a not very wrong resonance frequency for a tube if you claim that the open half is a pure inductance nd the closed half a pure capacitance.

 

Similarly, you could try to replace the belly of the Helmholtz resonator by an additional tube length of identical volume, but this is inaccurate when the added length adds much inductance, that is, when the diameters of the neck and belly differ much. Or you add a cylinder length that gives the same fundamental resonance frequency, but then the added volume won't match the belly's volume.

 

And in any case, the overtones differ completely between a bottle and a tube.