The bell-in-a-vacuum-jar sound experiment

[ozymandias]

Hi everyone,

I'm sure most of you are familiar with the "put a bell in a jar and pump the air out until you can't hear anything" experiment undergraduates are shown (e.g. see http://www.teralab.co.uk/Experiments/Bell_in_Vacuum/Bell_in_Vacuum_Page1.htm" ). Now, the "classic" explanation is that, as air is pumped out the sound dies away because it has no medium to travel in. At this point many people claim that way before that happens (i.e. way before the mean free path becomes ~ the wavelength) the impedance mismatch simply causes the sound to get reflected back into the jar. I was sure I could easily show this to be true, but alas! My hubris has backfired :). Please help me out here!

I've been trying to estimate really how much attenuation, say, a pressure of $$10^{-3}$$ atmospheres might cause (for such a pressure the mean free path is http://en.wikipedia.org/wiki/Mean_free_path" squared:

$$T^2 = 1 - \left(\frac{Z_2 - Z_1}{Z_2 + Z_1} \right)^2 \approx 0.004$$

where $$Z_1$$ is the acoustic impedance of the air inside the jar and $$Z_2$$ is the acoustic impedance of the air outside the jar. However, while this might seem impressive, sound behaves logarithmically; this decrease in transmitted power by a factor of 250 is equivalent to about 20-30 dB http://en.wikipedia.org/wiki/Sound_power_level" - not the entire picture, I think.

I'm guessing there is also an impedance mismatch between the sound source and the air in the jar - but I'm not sure how to estimate the resulting attenuation (compared to the case in which you'd have air at atmospheric pressure inside the jar). Would someone here perhaps have some insight to share?

 

[Baluncore]

Only with an impossibly perfect vacuum can there be no sound because there is no medium to transmit that sound. But that can never be.

I think your analysis is correct. There is no question that there are three impedance mismatches to consider. Two of those are internal, between the solid bell and the rarefied air, and between the rarefied air and the solid bell-jar. The external bell-jar to atmosphere mismatch will remain constant.

The speed of sound will be a function of temperature, so you must let that remain constant during the experiment to avoid being confounded by wavelength sensitivity.

The statement; "put a bell in a jar and pump the air out until you can't hear anything" sums it up. In reality there will always be some very quiet sound, but your ear will not be able to hear it above your blood flow. The lack of sound perceived is entirely due to attenuation by the two internal impedance mismatches.