Beer and Filters

I'm planning to show off my nice old Pickett N-515-T to some students in a few days time. I think I'll need a hook to get them interested and one sure way of attracting the attention of engineering students always used to be beer...

As is well known, empty (or part-empty) beer bottles, exhibit a nice acoustic second order resonance, analogous to the electrical resonance of an LC filter...

Reflecting the analogy, the resonant frequency is determined by exactly the same equation  I presented in the recent post on my beautiful old slide rule, where the 'capacitance' is the acoustic compliance of the volume of air in the body of the bottle and the 'inductance' is the acoustic mass of the 'slug' of air in the neck.

The equations for these parameters are...

in which rho is the mass density of air and c is the speed of sound.

Air is surprisingly 'heavy' stuff - such that its density at the ordinary temperatures and pressures in which you're likely reading these words is around 1.22 kg per cubic metre and the speed of sound is around 340 metres per second. V is the volume of air in the 'body' of the bottle, l is the length of the neck and S is the cross-sectional area of the neck.

There are a bunch of other details, which I'll not bore you with - such as the reason for saying 'acoustic' compliance above. Also, the length l needs an 'end correction' (an additional length), to account for the radiation of sound from the open end of the bottle and the cross sectional area of the neck is not constant in the Bud bottle pictured above, such that specifying S isn't easy!

Taking a look at the bottle through half-closed eyes (and noting it was specified on the label to hold 330mL), I came up with some very ROUGH ESTIMATES of V=295 e-6 cubic metres, l=0.09 metres and S=3.14 e-4 square metres.

These produce the following equally rough values for the 'capacitance' and the 'inductance' (don't worry about the units):

C = 2.09 e-9
L = 347

I tried to solve for the resonant frequency of (the air inside) the Bud bottle on the Pickett N-515-T (I wonder if that's been done before?) using exactly the same steps as previously described.

We start with the rough estimate on the Decimal Point Locator...

First, we find the capacitance of 2090 pF, since there isn't a nanoFarad scale (I've indicated 2000 pF by my green line and red circle)...

then we move the slide until the 2090pF lines up with 347 H on the slide. This is difficult to do, as 347 Henries would be one big mother of an inductor, so the scales didn't run that high, stopping at 100H.

Fortunately, there's clear space above 100 Henries and we can easily see how the logarithmic scale works so, extrapolating from the decade 10:100, we can 'imagine' the decade 100:1000 (which I've labelled in blue on the photo below) and position the slide approximately...

Then, we should be able to read off the resonant frequency from below the arrow...

it is around 200 Hz.

For more accuracy, we can multiply together the 'inductance' and 'capacitance' to find the LC product...

 
 to get 7.25 (e-7) and set the hairline of the cursor over this value on the cursor on the 'H' scale...

The resonant frequency is directly read from the D scale to be around 187 Hz.

Blowing over the beer bottle, in traditional bar-room style, whilst running a tuning app on the iPad, revealed an actual frequency of 196 Hz ...

so this model - and the rough estimates of sizes - was reasonably accurate.

The bottle is (as many will know) an instance of what is called a Helmholtz resonator, after German physician and physicist Hermann von Helmholtz. Hermann was way too dignified to blow over bottles in bar rooms - here's an image from my copy of his 'On the Sensations of Tone...', showing his equivalent of the Bud bottle...

He even arranged a nice little 'Guttapercha' tube to direct the blowing!

It is nice to see that the Pickett N-515-T is good for more than electronics. I wonder if many other folks have done acoustics on it - or taken it into the bar?

...-.- de m0xpd