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It is well known that if you blow horizontally on a bottle top it creates a sound. Pouring water to the bottle changes the pitch.

I have been doing experiments on the relation between the sound's main frequency (or rather the corresponding wavelength) and the vertical distance between the bottle top and the water level.

My (poor) theoretical thoughts were that by blowing on the top I create a standing wave exactly between the bottle top and the water surface, therefore the wavelength should be equal to the height of the air column.

Apparently, I was wrong, but not completely -- it's not $y=x$, but rather a linear relation $y=ax+b$. I have been conducting several experiments with different bottles, and the results are:

A 18cm high bottle gave me $a=5.5$ and $b=26.65$. A 21.7cm bottle gave me $a=7.94$ and $b=14.54$. A tall 32.5cm bottle resulted in $a=7.1$ and $b=34.21$. But my biggest surprise was to find a negative $b$, with a glass 19.3cm bottle (the rest were plastic), and $a=9.16$ and $b=-22.43$.

Is there a plausible explanation for this phenomenon?

(I didn't consider the actual shape of the bottle as I assumed a vertical standing wave will emerge. Most of the bottles are pretty constant in shape, except for the glass bottle which is closer to a cone. Looking at the signal itself one can clearly see a high energy peak at the pitch, and smaller peaks at the pitch's multiples. I noticed that the slopes are close to natural numbers or half thereof.)

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up vote 3 down vote accepted

I've linked a question that is closely related to yours.

If the bottle were behaving as a closed pipe you'd expect the wavelength to be about four times the air column height.

$$ \lambda \approx 4h $$

However a bottle of any significant width acts as a Helmholtz resonator instead, and the frequency is proportional to the square root of the volume:

$$ \lambda \propto \sqrt{V} $$

In your experiment the volume is related to a air column height by some complicated function related to the shape of the bottle, so it's not surprising a linear fit gives some odd results.

Response to comment:

The reason you get an oscillation is basically turbulence. Generally speaking fluids (gases and liquids) flow smoothly as long as the flow rates are slow, but as the flow rate increases the smooth flow becomes unstable. In the case of the bottle if you blow very gently you get no oscillation. You need to blow fast enough for the flow around the bottle opening to just start to become turbulent.

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Great answer, thank you! – ronash Apr 10 '13 at 8:08
Just a quick question: in the explanations for this phenomenon I see it is assumed that the airflow caused by an external force stops, and then the system oscillates. However, the sound is heard while you blow on the top, without stopping the airflow. Am I missing something? – ronash Apr 10 '13 at 9:08
@ronash: I've updated my answer to respond to your comment – John Rennie Apr 10 '13 at 15:28

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