Physics Puzzle Solution: No Sound in a Vacuum?
This is the solution to this puzzle.
The answer is no. The vacuum created in these experiments is usually not a vacuum at all; in fact, it’s nearly impossible to create a perfect vacuum. What usually happens is that the bell stops ringing once the density of the air inside the jar drops to about 1%-0.1% of its original value. Under such conditions, the drop in the bell’s volume is a result of its sound being reflected back into the jar. Sound is reflected because of the air pressure mismatch between the inside of the jar and the outside air, and between the bell and the air inside the jar; this is what an acoustician would call an acoustic impedance mismatch. It has nothing to do with there not being a medium to transmit the sound waves.
Let’s do some estimations. A bell’s ring is usually mid to high frequency - let’s take it to be 1000 Hz. This means its wavelength is about (speed of sound)/(1000 Hz) ~ 0.3 meters. For air to sustain this wavelength its density must be high enough such that the mean free path is much larger than 0.3 m. It isn’t too difficult to estimate the mean free path as a function of air pressure/density, but it’s even easier to consult Wikipedia on that matter. This reveals that, even at a pressure equal to 100 pascals (1000th of standard atmospheric air pressure, corresponding to an air density 0.1% as dense as “normal” air), the mean free path is ~ 100 micrometers = 0.1 mm, which is sufficient. Typical vacuum jar experiments don’t often go below this threshold.
How big is the drop in volume due to the impedance mismatch? There are two mismatches to take into account: one is between the bell and the air in the jar, and the other is between the air in the jar and the air outside (one should, in principle, also take into account the jar’s walls). Let’s estimate the second; the first is of similar magnitude.
The impedance of a gas is given by Z=ρv, where ρ is the density (in kg/m^3) and v is the speed of sound (in m/sec). The speed of sound is the same in both media (can you see why?). The density of the air inside the jar is about 1000th of that of the air outside the jar. The percentage of energy that gets transmitted from the air inside the jar to the air outside the jar is (see here for a derivation):
Don’t forget volume behaves logarithmically. The drop in SWL is:
The bell-air boundary provides a similar attenuation. This ~ 50 dB drop is enough to silence the bell’s ring long before the air becomes so thin it doesn’t allow for sound waves to travel through it.
