Helmholtz Resonator

I include this demonstration because many physicists get it wrong. The usual explanation for the pitch, produced by blowing over the top of a bottle, is the bottle length. This assumes the sound production is like an organ pipe. Middle C has a frequency of 261 Hz and a wavelength of 1.3 m or 4.1 feet. Bottles less than a foot in length produce notes lower than middle C. A two liter, 11 inch soda bottle produces a frequency of 105 Hz. Do you see the problem?

Rather the bottle is a Helmholtz resonator. The air mass in the neck of the bottle moves up and down according to Newton’s second law, when one blows across the top. The bulk of the air in the bottle serves as a spring.

Let the volume of the bottle be V, the length of the neck L, and the cross sectional area of the neck A. The average pressure in the bottle is atmospheric pressure Pa. The mass of the air in the neck is m = r A L, where r is the density of air. If the plug of air moves down a distance x, the volume in the bottle decreases to V- A x and the pressure increases to Pa + p. The oscillations are rapid enough that the compression process is adiabatic (following the adiabatic equation of state). Thus,

where gamma is the ratio of specific heats. Now consider Newton’s second law for the movement of the air mass in the bottle neck.

This is the equation for a harmonic oscillator with frequency

This can be written in terms of the speed of sound, c.

I used a 2 liter bottle, neck diameter 22 mm and length ~ 3 cm. This gives a frequency, f = 132 Hz. This is reasonably close to my measured value given the uncertainty of L. The wavelength is 2.5 m, much larger than the 11 inch length of the bottle.

To fiddle with A, I put a piece of tape across the bottle top. When I blow from the tape side, a much higher pitch is produced. Ill have to take more measurements.