# Resonant frequincy of capped cylinders

## Main Question or Discussion Point

I'm looking for a formula to calculate the resonant frequency of a capped cylinder. I have found a link to a formula but was surprised to see that cylinder radius is not taken into account when making the calculation.

It was my understanding that the larger the radius of a cylinder the lower the note produced, is that correct?

How dose cylinder radius relate to the note produced eg if I had five cylinders of equal length but each one being twice the radius of the other what effect would this have on the sound produced?

Here is a link to the formula I mentioned

hyperphysics.phy-astr.gsu.edu/hbase/waves/clocol.html#c1

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f95toli
Gold Member
I'm looking for a formula to calculate the resonant frequency of a capped cylinder. I have found a link to a formula but was surprised to see that cylinder radius is not taken into account when making the calculation.

It was my understanding that the larger the radius of a cylinder the lower the note produced, is that correct?
No, the radius might affect things like the quality factor (i.e the width of the resonance) but the resonance frequency itself (i.e. the centre frequency) only depends on the length.

There are of course OTHER resonances in a capped cylinder that depend on the radius etc (there are usually plenty of resonance frequencies in real 3D objects) but they in turn do not depend on the length and occur at other frequencies. Moreover, they are usually not exited in a long tube and you rarely need to take them into account.

Consider the following experiment:

An empty paint can (capped cylinder) is half filled with water and positioned over a tap so that the tap is over the exact centre of the can.
The tap is now finely adjusted so that a single drop of water drips from the tap. The water drop hits the surface of the water in the can and produces a ripple. The ripple travels out from the centre of the can and rebound off the walls of the can only to converge back to the centre of the can again.
The tap is so adjusted that as the wave converges at the centre point another droplet hits the water surface. In this way resonance is set up. It will be found that the radius of the can determines the resonant frequency.

f95toli
Gold Member
Yes, but what you are describing is not the resonance frequency of a capped tube; it is a resonance of a disc/membrane (which just happens to be one end of a tube); i.e. essentially a 2D problem since in your example the resonance frequency does not depend on the length.
As I wrote above: There are MANY different resonances (modes) in a real capped tube; including some exotic ones like whispering gallery modes.

This problem is usually solved in courses in Fourier analysis when the method of separation of variables is introduced; it is relatively straightforward in cylindrical coordinates since there is axial symmetry and -depending on the initial conditions- you will end up with a solution involving products of Bessel functions, cos and sin functions.

Unfortunately I can't find a good link, but you should be able to find the answer in any book on Fourier analysis.

Thanks f95toli, I am looking to machine cavitys to resonate at ultrasonic frequencies. The cavitys will therefore be relatively short in length, this is why I was thinking about the radius of the cavitys. When a cavity is relatively short I'm thinking that sort of resonance I mentioned before might be setup and therefore cavity radius may become an important factor?