Frequency of a Helmholtz resonator

[djembeweaver]

I've never been quite sure how the dimensions of a djembe determine the frequency of the bass note, and I've often noticed that very small drums have surprisingly low bass notes, so I've been doing a bit of research and came across this website explaining helmholtz resonance:

http://www.phys.unsw.edu.au/jw/Helmholtz.html

It gives the formula for the frequency of a (mathematically perfect) helmholtz resonator as (deep breath):

The speed of sound divided by 2pi times the square root of the cross sectional area of the aperture divided by the length of the neck times the volume of the container.

Sorry I couldn't copy the formula over.

Given that the speed of sound doesn't change very much and pi is a constant that means that the frequency is determined by the ratio of the cross sectional area of the aperture (the size of the hole if you like) to the volume (size of the bowl) times the length of the neck (stem of the djembe).

This means that if you took a full-size djembe and scaled it down exactly the bass note should not change frequency!

Obviously it also means that a smaller hole should produce a lower frequency etc.

Just thought I'd share that. It answered a few questions for me.

Does anyone have any examples of this in practice (particularly people who have actually carved shells)?

[michi]

I posted some results back in March here. But it didn't occur to me to compare the bass of different-size shells.

I have a whole bunch of drums kicking around here at the moment (about 20). I'll pick a few with different sizes over the next few days, record the bass, and run a spectrum analysis. It'll be interesting to find out what they all do, no doubt

Michi.

[djembeweaver]

Awesome. Thanks Michi!

[bkidd]

djembeweaver wrote:
The speed of light divided by 2pi times the square root of the cross sectional area of the aperture divided by the length of the neck times the volume of the container.

Given that the speed of light doesn't change very much and pi is a constant that means that the frequency is determined by the ratio of the cross sectional area of the aperture (the size of the hole if you like) to the volume (size of the bowl) times the length of the neck (stem of the djembe).

I think you mean the speed of sound, rather than light. Nonetheless, this is a great example of acoustics playing into djembe design.

Best,
-Brian

[djembeweaver]

Whoops - yes, how embarassing!

My only excuse is that as I was translating the formula into words I looked at "c" and just wrote light instead of sound.

[michi]

We'll forgive you. The speed of sound is pretty constant too, at least in those places where people play djembe

Michi.

[djembeweaver]

Duly corrected to avoid misinformation.

Nothing to see here guv'

[bkidd]

Michi wrote:
We'll forgive you. The speed of sound is pretty constant too, at least in those places where people play djembe

Maybe I'll start up a new sport... underwater djembe playing

[djembeweaver]

Michi wrote:
We'll forgive you. The speed of sound is pretty constant too, at least in those places where people play djembe

Maybe I'll start up a new sport... underwater djembe playing

Even better would be vacuum drumming. No worries about annoying the neighbours then. Let me see....just need to hermetically seal my living room.....

[drtom]

Michi wrote:

I'll pick a few with different sizes over the next few days, record the bass, and run a spectrum analysis

I'd be really interested to see the results Michi. Please do let us know what you come up with.

drtom

[michi]

Sorry I couldn't copy the formula over.

Here are the diagram and the formula:

Schematic of a Helmholtz resonator

Helmholtz2.GIF (2.18 KiB) Viewed 140 times

Resonant frequency of a Helmholtz resonator

Helmholtz7.GIF (1.01 KiB) Viewed 140 times

Given that the speed of sound doesn't change very much and pi is a constant that means that the frequency is determined by the ratio of the cross sectional area of the aperture (the size of the hole if you like) to the volume (size of the bowl) times the length of the neck (stem of the djembe).

That's right. It's purely the size of the bowl, the diameter of the neck, and the length of the stem that determine the frequency.

This means that if you took a full-size djembe and scaled it down exactly the bass note should not change frequency!

No, not really. That's because the volume of the bowl increases as the cube of the linear dimensions, so the term in the denominator gets disproportionately larger as the drum is scaled up.

Let's try this…

Here is an image of Helmholtz resonator with proportions (very roughly) like a djembe:

Djembe shaped (more or less) Helmholtz resonator

Helmholtz.jpg (26.18 KiB) Viewed 129 times

Let's assume more-or-less normal djembe size: sphere diameter 0.3m, stem height: 0.3m, stem diameter: 0.075m. This gives a sphere volume of 0.0045m³ and a hole area of 0.0176m². Plugging these numbers into the formula, we get a resonance frequency of 110Hz.

Now let's halve the dimensions, so we have sphere diameter 0.15m, stem height 0.15m, stem diameter 0.0375m. This gives a sphere volume of 0.00177m³ and a hole area of 0.00442m², for a resonance frequency of 220Hz.

So, if you halve the size of the drum, you get double the bass frequency.

Now for the experiment

Here is an image of the drums I used. I've labelled each drum with its bass frequency. (All of these were skinned by me, except number 2 & 3.)

Drums used to measure bass frequency

10 Drums - small.jpg (166.38 KiB) Viewed 140 times

You can look at a higher-resolution image here.

The drums are 60cm tall (with minor variation), except for the first two, which are 50cm tall. The bougarabou is a little taller, around 63cm, and the Jina djembe is a little shorter, about 58cm.

Left to right:

1. 90Hz: The first drum I ever owned. (I also built it. The skin on it is still the same skin that I fitted eight years ago. ) 50cm tall.
2. 82Hz: Cheap Indo drum, lathe turned. 50cm tall.
3. 70Hz: Cheap Indo drum, lathe turned.
4. 66Hz: Indo drum, lathe turned.
5. 63Hz: Indo drum, lathe turned.
6. 80Hz: Bougarabou, Iroko.
7. 65Hz: Mali djembe, Lenke.
8. 64Hz: Mali djembe, Gueni (Jina djembe).
9. 72Hz: Guinea djembe, Lenke (Drumskull).
10. 68Hz: Guinea djembe, Djalla (M.K. Signature Series).

As you would expect, the three small drums have higher bass, and all the others (except for the bougarabou) are fairly closely clustered.

The bougarabou's frequency is so high because S is large, and L is small, making the term under the square root larger.

So, after all this, we now have scientific proof that size matters

Cheers,

Michi.

[djembeweaver]

Great work. Thanks for going to all that trouble (to prove what you already knew!)

No, not really. That's because the volume of the bowl increases as the cube of the linear dimensions, so the term in the denominator gets disproportionately larger as the drum is scaled up

Bang on. That will teach me not to make assumptions before working through actual numbers!

Let's assume more-or-less normal djembe size: sphere diameter 0.3m, stem height: 0.3m, stem diameter: 0.075m. This gives a sphere volume of 0.0045m³ and a hole area of 0.0176m². Plugging these numbers into the formula, we get a resonance frequency of 110Hz

Hmmn...using the same dimensions I get a sphere volume of 0.0141m³ and a hole area of 0.0044m². That gives a frequency of 55 Hz.

Now let's halve the dimensions, so we have sphere diameter 0.15m, stem height 0.15m, stem diameter 0.0375m. This gives a sphere volume of 0.00177m³ and a hole area of 0.00442m², for a resonance frequency of 220Hz

I get a sphere volume of 0.00177m³ and a hole area of 0.0011m² and a frequency of 110 Hz.

That's a long winded way of saying I agree with your conclusion but one of us has got the numbers slightly wrong (probably me, but I've plugged it all into a spreadsheet and error-checked every cell twice)

Anyway, I think you've clearly demonstrated that size matters

[michi]

Hmmn...using the same dimensions I get a sphere volume of 0.0141m³ and a hole area of 0.0044m². That gives a frequency of 55 Hz.
[…]
That's a long winded way of saying I agree with your conclusion but one of us has got the numbers slightly wrong (probably me, but I've plugged it all into a spreadsheet and error-checked every cell twice)

Right you are! That's because I'm too dang stupid to know the difference between the radius and the diameter of a hole! I accidentally used the hole diameter instead of the radius, so I ended up with frequencies double what they should be… Your calculations were dead right!

But, basically, actual numbers don't really matter here. If the three (linear) dimensions for the radius of the sphere, the radius of the hole, and the length of the stem are a, b, and c, and we substitute in the formulas for the volume of a sphere and the area of a circle, we get:

Helmholtz formula including formula for volume of sphere and area of opening

F1.gif (1.63 KiB) Viewed 118 times

To work out what happens when we double the linear dimensions, we need to evaluate the following ratio:

Ratio of two drums differing in size by a factor of 2

F2.gif (1.94 KiB) Viewed 118 times

That's a drum at the top that is twice the size of the one at the bottom.

This then simplifies as follows:

Simplifying the ratio

F3.gif (4.9 KiB) Viewed 118 times

And that says that, if I double the size of the drum, I get a bass at half the frequency (one octave lower). Or, if you prefer, if I halve the size the of the drum, I get a bass at twice the frequency (one octave higher).

Anyway, I think you've clearly demonstrated that size matters

I hope so!

Michi.

[djembeweaver]

Your calculations were dead right!

Right on a point of pedantry...story of my life!

And that says that, if I double the size of the drum, I get a bass at half the frequency (one octave lower). Or, if you prefer, if I halve the size the of the drum, I get a bass at twice the frequency (one octave higher).

Agreed. Phew - that was hard work

To link this with the other thread, are we saying that the size of the waves determines the bass, but the motion of the ocean determines slaps and tones?

[michi]

To link this with the other thread, are we saying that the size of the waves determines the bass, but the motion of the ocean determines slaps and tones?

Wow! That's getting too metaphysical for me

It's bit like asking "When a man speaks, and there is no woman to hear him, is he still wrong?"

Michi.