RE: Cars and the Pop Bottle Effect

Dear Tom,

One of your callers a couple months ago had a question about why her car shakes when she has the windows up and the sun roof open.

This reminded me of the 1965 VW bug I had in high school. A favorite pastime was to open the sun roof and close the windows at highways speeds, which would make the whole car shake violently with a low-frequency vibration. People who hadn’t experienced this before were usually very impressed.

My friend Del Koch and I dubbed it the “Pop Bottle Effect,” since we assumed that what we were experiencing was the same thing that happens when you blow over the top of a pop bottle, only in this case the whistle was very low frequency and we happened to be inside the pop bottle, driving it though the air.

Being a physicist, I’ve thought about this over the years and wondered if I could show that we were right—that my VW bug really was acting like a giant pop bottle. In particular, I wondered if I could show that the same equation that determines the pitch of a whistle from a pop bottle, when applied to something the size of my VW, would give a whistle frequency of 5 to 10 hertz, which is what I estimated my VW gave.

When I heard your recent caller, I decided it was time to get to the bottom of this.

What’s happening, of course, is that the car is acting as a resonant cavity, but the question is how to figure out what frequency the cavity is resonant at. The case we all learned in school was the simple case of a organ pipe (a long, narrow tube), but that didn’t seem to fit. So I got on the web to see what determines the resonant frequency of a more complicated shaped cavity.

To my surprise, I couldn’t find the answer. So I decided to just do the experiment.

I collected four bottles—all of different shapes and sizes—ranging from a small plastic cough medicine bottle to a gallon jug that held bleach. Then I took them over to my musician friend Ben Rudnick’s house. I’d blow across the top of them and he and his daughter would match the resulting tone with a note on the piano. I would write down the note, which allowed me to figure out the frequency of the whistle. By partially filling the bottles with water I could change their volume, and I collected 10 data points of resonant frequency vs. cavity volume.

Once I had the data, I first wanted to see if the bottles were just acting like an organ pipe; if that were the case the resonant frequency would just depend on the length of the bottle, that is, the distance from the mouth of the bottle to the bottom, or the distance from the mouth to the top of the water I’d put into the bottle. So I plotted the 10 resonant frequencies against the length of the bottle. The result is shown in Figure 1, where each of the bottles is represented by a different symbol, and multiple points with the same symbol correspond to a bottle with different amounts of water in it.

The plot clearly shows that the points don’t fall on a curve, so the resonant frequency depends on something other than just this distance. This means that the bottles are not acting like simple organ pipes.

Next I plotted the resonant frequencies against the volume of air in the bottles. That plot is shown in Figure 2.

The curve through the points in Figure 2 is a fit of the data to a power law. The equation describing that curve relates the resonant frequency f to the volume V of the bottle:

where f is in hertz.

What this means is that the resonant frequency scales as roughly one over the square root of the volume. There is probably a simple reason why this is true, but I haven’t figured it out yet.

We can now use this equation and see what resonant frequency it gives for the volume of a VW bug. With the seats, people’s bodies, etc., inside the car, it’s not completely clear what volume to use, but it’s somewhere in the range of 30 to 50 cubic feet (eg, 3ft x 3ft x 5ft = 45 ft³). Figure 3 shows a plot of the frequency-volume equation for volumes in this range.

What this plot shows is that a volume of 30 to 50 cubic feet gives a resonance frequency of 5 to 6 hertz, which is just what I had observed in my high school days.

So Del and I were apparently right, and this low-pitch whistle really is the Pop Bottle Effect.

Regards,

David Wright

2 Windermere Lane

Arlington, MA 02476

dcwright@mit.edu