How high altitude/low pressure could you could hear a gong right next to you?

Question

"There is no sound in space". But how thin can air be and musical instruments still be heard?

The absolute pressure limit of sound (of a given wavelength) occurs when the wavelength is less than the mean free path of the molecules. Setting the frequency to 20 Hz (human "lower limit") gives us about 160 km. However, this question is about the limits of musical instruments vs human ears: the actual altitude could be much lower than 160 km. It is a "practical physics" question along the lines of power-line sensations and of course must be answered approximately.

Suppose you are standing 1 meter away from a gong when it is hit with full force. You are on a separate platform than the gong; the only vibration/sound is perceived through the air. You cannot see the gong being struck (to avoid the placebo effect). You are a facultative anaerobe but otherwise are an unmodified human.

Considerations

A gong is about 90 dB when played vigorously at sea level. A single strong hit may reach 105 dB or so from a 1 meter distance.

Human hearing sensitivity varies with frequency. The "threshold" of zero dB (at our most sensitive frequency) is slightly generous, 5-10 dB is more typical for healthy ears. Here we define "threshold" as hearing it even if you don't expect it, assuming you are not deeply lost in thought. This is a bit higher than two alternative forced choice and may add another 5-10 dB. Any instrument has multiple frequencies and loudness presumably is a weighted sum of how much energy and how sensitive our ears are at each frequency.

There are three effects that hamper hearing at extreme altitudes:

The direct effect: As the air thins, the gong will transmit less energy into the air.

Impedance mismatch: The thinner air means that there is less mass pushing on the eardrum. Compared to Mars-density air, an eardrum is almost a solid wall. Most incoming sound energy would be reflected and not get to the inner ear.

Damping: More and more sound is attenuated (damped out, as energy gets converted to heat) as we get higher and the mean free path increases. The gong may sound muffled if high frequencies are cut off, although it's a low frequency instrument so should be less affected. Over a 1 meter distance, I am not sure if attenuation will be relevant to the perception limits or if the other two effects will kick in first.

Given these considerations, what is the approximate altitude limit for hearing a gong?

Answer 1

If we assume the inverse square law for sound intensity $I(r)$: $$ I(r)=\frac{I_0}{r^2}, $$ where $r$ is distance in meters and $I_0$ is intensity at 1 meter distance, then doubling the distance will cause the intensity to drop by approximately 6 dB: $$ I_{dB}(r)=10\log_{10}(I(r))=10\log_{10}I_0-20\log_{10}(r)=I_{dB}(1)-20\log_{10}(r), $$ $$ I_{dB}(2)=I_{dB}(1)-20\log_{10}(2)\approx I_{dB}(1)-6 $$

If $I_{dB}(r)=10$, $I_{dB}(1)=90$, then: $$ 20 \log_{10}(r)=I_{dB}(1)-I_{dB}(r)=80 $$ $$ \log_{10}(r)=4 $$ $$ r=10^4m=10km $$ Sound attenuation will decrease this distance.