Question about Helmholtz resonance

[Sylvester McBean]

Perhaps my question has to do with Helmholtz resonance, perhaps not. That's why I'm here. ;-) Here's my question:

Say you have a large steel oil drum that is half full of water. If you bang on the side of the drum towards the top with another metallic object, what exactly is making the sound you hear? I understand that it would be a complex sound, but is it primarily the resonance frequency of the steel drum or the air inside the steel drum that is causing the sound to have a particular frequency/pitch? I believe that the less water in the drum, the lower the pitch of the resulting sound, and vise versa, is that correct? Would the amount of water in the drum and the frequency/pitch of the sound be expected to have a linear relationship?

Originally I thought it was the resonance frequency of the steel that was causing the sound, which you would expect to increase as the water in the drum increases, since the water has a dampening effect on the vibration of the steel wall. But then I read about Helmholtz resonance and I second guessed myself. What I don't know is whether or not Helmholtz resonance applies to 'closed' systems like our oil drum.

Any help is much appreciated, thank you!

 

[tech99]

I think you are hearing the steel because you are not exciting the correct mode in the air for the Helmholtz resonance.. The Helmholtz resonator requires a narrow neck, and the sound pulsates in and out. The frequency is determined by the mass of air in the neck and the springiness of the air volume in the container. It should give a pure and lingering tone. I am guessing that a container of the size you are considering would have a Helmholtz resonance in the low audio range, not the higher frequency bell-like sound of steel..

 

[sophiecentaur]

The Helmholtz resonance would be due to the air inside the drum moving in and out of the 'port'. The natural frequency would be a function of the size of hole and the volume of air. If you want to excite that mode of resonance you can slap your hand over the hole and listen to the low pitched 'thump'. Alternatively you could try to hum a low note and you may get it to resonate. Increasing the water level (hence decreasing the volume of air) will increase this frequency. It's the design basis of many sub-woofer loudspeakers as it has the equivalent resonance of a very long organ pipe. It's ok initially to assume that the steel is rigid enough to ignore its contribution to that mode of oscillation.
Hitting the drum with a hammer will excite the steel and it will probably 'clang!' The water will affect the pitch by damping the waves along the sides of the cylinder but the way the steel oscillates is a much more complex matter.

 

[Sylvester McBean]

Thanks for the replies. What is much more complex about the way steel resonates? Can you elaborate a bit on that? Also, would the relationship between the water in the drum and the pitch be linear? I would think it would...

 

[sophiecentaur]

Thanks for the replies. What is much more complex about the way steel resonates? Can you elaborate a bit on that? Also, would the relationship between the water in the drum and the pitch be linear? I would think it would...

The Helmholtz resonator is very small in relation to the wavelength of the resonant frequency so it can be pretty accurately modeled as three 'lumped components' - the electrical equivalent is a single Capacitance, a single Inductance and some resistance (loss). The walls and the water do not interact significantly with the air waves so you can ignore them.

The drum is another matter. There are at least three 'components' involved. The lid is a circular membrane which has many different modes of oscillation (like a drum membrane), the sides are a cylindrical membrane which also can support several modes. Also, the water, being fairly dense, loads the resonator in a complicated way. I found this link which describes how the vibrations are formed in a part filled wineglass. Section III (Theory) starts off by saying it's complex!.

If you wanted to examine the linearity between water level and frequency, you could take a tall jam jar (I doubt that you could easily get hold of a thick walled metal cylinder as easily as a glass one) and do a simple kitchen tabletop experiment. It needn't be full of wine; water would do.

PS It might be worth while getting @OmCheeto involved with this as he is a champion home experimenter and he may well be fond of wine, too.

 

[OmCheeto]

...
PS It might be worth while getting @OmCheeto involved with this as he is a champion home experimenter and he may well be fond of wine, too.

Thanks for the mention, but it appears that the original question has already been answered.
Refining the question further, strikes me as a bit of a worm-hole type of problem.

How I perceive the question ending up:

We have 3 different mediums: air, liquid, and solid
Each has different speeds of sound.
The configuration of the solid can take infinite forms.
The level of the liquid varies from zero to full.
What is the resonant frequency?​

Nope. Not even going to try and help on this one.

I found this link which describes how the vibrations are formed in a part filled wineglass. Section III (Theory) starts off by saying it's complex!.

Even before I looked at your (to me, incomprehensible) link, I looked at the following: Singing Glasses [SciAm]
Which looks like the same experiment, only with far fewer big words.
I'm actually considering doing the Sci-Am experiment, somewhat modified, as I've noticed over the years, when stirring my coffee in the morning, after adding sugar, the frequency changes.

Not sure if it's a real, or just a "hurry up!" perceived time dilation "I need my coffee now!" effect.

 

[sophiecentaur]

Each has different speeds of sound.

It's worse than that because the audible waves along the steel shell would be mainly transverse waves (as in a bell) and the various thicknesses throughout would mean different speeds for lid and sides and the drum could have reinforcing ridges round the cylinder too.

Even before I looked at your (to me, incomprehensible) link

Not much better for me either - I just put it in for devilishness.

 

[OmCheeto]

...
Not much better for me either - I just put it in for devilishness.

Being a bit less devilish, I left out the two following videos that I watched, trying to figure this problem out, as, although incredibly fascinating, I didn't think they captured the essence of this problem:


Earthquakes, Circles and Spheres - Numberphile


Coffee Cup Vibrations - Numberphile

In any event, I have never not enjoyed one of Tadashi Tokieda's explanations, of anything.

 

[OmCheeto]

mainly transverse waves

Actually, this comment is why I decided to include the above videos.
I had never looked into the difference between "P" & "S" waves before.
But I decided that they kind of describe how much more complicated "things" are, than the original problem might originally imply.

 

[sophiecentaur]

Actually, this comment is why I decided to include the above videos.
I had never looked into the difference between "P" & "S" waves before.
But I decided that they kind of describe how much more complicated "things" are, than the original problem might originally imply.

Harder still. The P and S waves are descriptions of what goes on, deep inside a substance (bulk waves). For a thin sheet you are dealing with surface waves which are like the ones which emanate from the epicentre of an earthquake and are neither P nor S. Those devils have H and V components (as ocean waves) and are responsible for shaking buildings down. The wave speed of these is a lot slower than the bulk P and S modes but they die out over a short distance. It's a bit like the difference between waves in the steel in a guitar string and the way the string itself moves.

 

[spareine]

Would the amount of water in the drum and the frequency/pitch of the sound be expected to have a linear relationship?

For vessels that do not have an upper lid, like wine glasses, the diagram of the frequency as a function of the height of the water is shown below. The fuller the glass, the lower the tone. The curved lines are approximately linear at both ends. The tangent line from the left intercepts the asymptote from the right at approximately d/H = 1/4. A.P. French (In vino veritas) derived a formula.

Blue lines: frequency of wine glasses as a function of the distance between the water level and the rim of the glass. H is height of the glass. Orange lines: same data, but (f₀/f)² on the vertical axis. f₀ is the frequency of the empty glass.

 

[OmCheeto]

...
A.P. French (In vino veritas) derived a formula.
...

Got to the end of that pdf, and realized how hard this problem is:

ps. Ok. It's a different article. But I have equal chances of grasping the complexities of either subject.

 

[sophiecentaur]

All along I have been assuming that an “oil drum” would have a lid over the top with a smallish hole in the top. That is what would constitute a Helmholtz Resonator. The frequency for that would be pretty much proportional to the volume of air inside. That would be a linear relationship with the amount of water.
That would be much simpler than the clanging sound of the ringing metal.

 

[spareine]

The OP asked if you bang on the side of a drum, what is causing the sound to have a particular frequency/pitch, and he mentioned the vibration of the steel wall and the vibration of the air as two possible explanations. I chose the first option. One consideration for me is that a bang on the side of a drum sounds like vibrating steel, not like resonating air. Another reason is that the OP specified that he banged on the side, as if he had noticed that a bang on the top lid produced another sound. Another sound would mean the sound was produced by a vibration of the steel wall. That is similar to a steel paper bin here in my room: a bang on the bottom sounds different than a bang on the side.

Below the sound spectrum, obtained with a smartphone, of banging on the side of the bin, while filling the bin with water. The frequency of several components decreased while the water was rising, as is typical for a vibration of the steel wall. In the second recording, labeled "rim unable to vibrate", I put my hand and arm on the rim to prevent it from vibrating. That was my poor man's imitation of a top lid. Although the sound quality was different, and the intensity of the components was different, the frequency decrease of several components was present again.

As the OP has access to an oil drum, he might check if a bang on the top lid sounds different than a bang on the side, and he might examine if the pitch decreases when the water rises. The change of pitch is audible, recording a spectrum is not required.

 

[sophiecentaur]

One consideration for me is that a bang on the side of a drum sounds like vibrating steel, not like resonating air.

I agree. The problem was in the thread title and nothing about wineglasses or bell-like resonances involves a simple Helmholtz resonance so perhaps that should have been totally sorted out earlier on.