thermodynamics - At what altitude would the air be too thin to carry a sound wave?

A related question When does an aerobraking space craft create a sonic boom? has spawned a couple of answers, but so far no compelling answers.

It is a common belief that in space there is no sound, How far do you have to be from Earth to be “in space”? does not have a definitive answer. But the Kármán line, at an altitude of 100 km (62 mi) above sea level is often given, it seems unlike that this is the exact altitude that air becomes too thin to carry a sound wave.

The top voted answer at What is the minimum pressure of a medium for which a sound wave can exist? seems to imply there is someplace where sound would fall below human thresholds. But does not relate it to an elevation.

At what altitude would the air be too thin to carry a sound wave?

Answer

First, I am going to provide a little background on equivalent pressures at different altitudes from Earth's surface.

Troposphere to Mesosphere
At sea level, the neutral atmosphere of Earth has a pressure of ~$10^{5}$ Pa (or ~1000 mbars).

The below image from https://en.wikipedia.org/wiki/File:Comparison_US_standard_atmosphere_1962.svg shows the broad range of temperatures/pressures of Earth's atmosphere.

Ionosphere
The region where the atmosphere transitions from mostly neutral to mostly ionized gas (called a plasma) is called the ionosphere. The altitudes defining this region vary (due to solar variability), but are generally defined as ranging from ~60-1000 km. The free electron number density in the ionosphere varies greatly from ~$10^{3} - 10^{6}$ # $cm^{-3}$ (or number of particles per cubic centimeter). The temperature varies from few 100 K to ~1500 K. Thus, if we treated it like an ideal gas the thermal pressure of the charged particles would range from a few $10^{-12}$ Pa to few $10^{-8}$ Pa.

Thus, the ratio of sea level pressure to the plasma constituents would be ~$10^{13} - 10^{17}$.

Plasmasphere

The region immediately surrounding the ionosphere is called the plasmasphere, which can extend to altitudes as low as a few $R_{E}$ up to ~6 $R_{E}$. The density ranges from several 100 # $cm^{-3}$ down to ~10 # $cm^{-3}$ and temperatures vary greatly, from ~6000-35,000 K. Again, these ranges correspond to thermal pressures of $10^{-13}$ Pa to few $10^{-11}$ Pa.

Thus, the ratio of sea level pressure to the plasma constituents would be ~$10^{16} - 10^{18}$.

Outer Magnetosphere
The "best" vacuum that we can easily access is the Earth's outer magnetosphere, which has a density ranging from ~0.01-1.0 particles per cubic centimeter. The temperatures in the outer magnetosphere can vary greatly from ~$10^{5}$ K to greater than $10^{9}$ K (i.e., if one converts radiation belt particle energies, e.g., 100s of keV, to an equivalent temperature). Thus, the range of equivalent ideal gas thermal pressures would be few $10^{-14}$ Pa to a few $10^{-8}$ Pa.

Thus, the ratio of sea level pressure to the plasma constituents would be ~$10^{13} - 10^{19}$.

Solar Wind
The solar wind - the supersonic flow of plasma from the sun's upper atmosphere - has densities and temperatures ranging from ~0.5-50 # $cm^{-3}$ and ~$10^{4}$ K up to ~$10^{6}$ K, respectively (for references, see https://physics.stackexchange.com/a/179057/59023). Thus, the range of equivalent ideal gas thermal pressures would be few ~$10^{-14} - 10^{-10}$ Pa.

Thus, the ratio of sea level pressure to the plasma constituents would be ~$10^{15} - 10^{19}$.

At what altitude would the air be too thin to carry a sound wave?

Short Answer
For all practical purposes, there are no regions of space completely devoid of some kind of sound.

Long/Detailed Answer
Interestingly, one can have the traditional sound wave (i.e., a longitudinal oscillation mediated by gas particle collisions) propagate into the upper atmosphere and even into the ionosphere. The point where such a sound wave would experience strong damping is where the collisional mean free path becomes too large to support the oscillations, i.e., this would occur when the average time between collisions becomes comparable to the wave frequency. Thus, the oscillations would have no restoring force and would damp out (for an electromagnetic analogy, see evanescent waves).

It is a common belief that in space there is no sound...?

No, there are sound waves that start in space can propagate in almost zero pressures, well, less than $10^{-14}$ Pa is as close to vacuum as one needs since the best vacuums we can produce in a lab is ~$10^{-11}$ Pa.

There are multiple sound waves in space, including ion acoustic waves and magnetosonic waves. Ion acoustic waves have been seen all the way out at Saturn, where the solar wind density and temperature can be as low as ~0.1 # $cm^{-3}$ and less than ~$10^{4}$ K, or thermal pressures below $10^{-14}$ Pa (note that the ram or dynamic pressure is generally ~2-4 orders of magnitude higher owing to the high speed of the solar wind). These modes have been observed as far out as Neptune, throughout the solar wind, and as in close as ~0.3 AU.

There is no reason to expect that such modes would not exist in the interstellar medium, where densities and temperatures can be as low as ~0.1 # $cm^{-3}$ and ~$10^{3}$ K, corresponding to $10^{-15}$ Pa.

The intra galaxy cluster medium is even more tenuous but much hotter, with densities and temperatures as low as ~$10^{-4}$ # $cm^{-3}$ and ~$10^{7}$ K, corresponding to $10^{-14}$ Pa (e.g., see arXiv e-print 1406.4410). Again, the ubiquity of ion acoustic waves in the interplanetary medium suggests we should expect these in nearly all regions of space.

Let me ask the question in a slightly different manner.

At what altitude would I no longer be able to hear a 100 dB source (ignoring suffication)?

The intensity of sound decreases as $I\left( r \right) \propto r^{-2}$ while sound pressure decreases as $P\left( r \right) \propto r^{-1}$. The hearing threshold is a function of frequency, because the human ear does not have a flat frequency response, but it is generally accepted as being ~20 $\mu$Pa for 1 atmosphere and 25$^{\circ} C$ at 1000 Hz. The sound pressure level (measured in dB) is given by:

$$L_{p}\left( r \right) = 20 \ \log_{10} \left( \frac{ P\left( r \right) }{ P_{o} } \right)$$ where we set $P_{o}$ ~ 20 $\mu$Pa. Then a 100 dB source corresponds to ~2 Pa at the source. This would drop to $P_{o}$ at a distance of ~$10^{5}$ m, ignoring any acoustic impedance or losses and assuming the pressure and temperature are the same as the reference parameters for $P_{o}$.

The reference sound intensity, $I_{o}$, depends upon the characteristic acoustic impedance, $z_{o}$, as $I_{o} = P_{o}^{2}/z_{o}$. We know that $z_{o} = \rho \ C_{s}$, where $\rho$ is the mass density and $C_{s}$ is the speed of sound. We can model $\rho = \rho \left( h \right)$ using a known atmospheric scale height and an exponential decay (which reproduces the orange line the figure above) and take a set of values from the blue in the figure above for $C_{s}$ (see table below). Then we find that $z_{o}$ ranges from ~0.003-416 Pa s/m from 0-100 km altitude. If we use the human hearing threshold for $P_{o}$, then $I_{o}$ ranges from ~$10^{-13} - 10^{-7}$ W $m^{2}$.

Altitude [km] | Cs [m/s] | Density [kg/m^3] | z_o [Pa s/m] | I_o [W/m^2]
-------------------------------------------------------------------------
      10      |   300.0  |     3.736e-01    |   1.121e+02  | 3.569e-12
      20      |   295.0  |     1.152e-01    |   3.399e+01  | 1.177e-11
      30      |   301.3  |     3.553e-02    |   1.071e+01  | 3.736e-11
      40      |   316.0  |     1.096e-02    |   3.462e+00  | 1.155e-10
      50      |   329.0  |     3.378e-03    |   1.112e+00  | 3.599e-10
      60      |   320.0  |     1.042e-03    |   3.334e-01  | 1.200e-09
      70      |   298.0  |     3.213e-04    |   9.573e-02  | 4.178e-09

      80      |   276.0  |     9.906e-05    |   2.735e-02  | 1.463e-08
      90      |   276.0  |     3.055e-05    |   8.431e-03  | 4.744e-08
     100      |   287.0  |     9.420e-06    |   2.703e-03  | 1.480e-07

Since $I_{o}$ increases as we increase altitude, then the intensity of our source would have to increase as well to maintain its initial $L_{o}$ = 100 dB level (i.e., $I_{src}\left( h \right) = I_{o}\left( h \right) 10^{L_{o}/10}$). The intensity at the source, $I_{src}$, then ranges from ~0.01-1500 W $m^{2}$.

Let's assume we use the same intensity at sea level and bring the speaker up in altitude, then the intensity level at the source drops with increasing altitude as:

$$L_{i,src}\left( h \right) = 10 \ \log_{10} \left( \frac{ I_{src}\left( 0 \right) }{ I_{o}\left( h \right) } \right)$$ Then $L_{i,src}$ varies from 100 dB at sea level to ~94 dB by 10 km, ~79 dB by 50 km, and ~48 dB by 100 km.

We estimate the intensity level at a given distance away from the source as:

$$L_{r}\left( h, r \right) = L_{i,src}\left( h \right) + 20 \ \log_{10} \left( \frac{ 1 }{ r } \right)$$ where we have used 1 m as a normalizing length defining at the source. In the following, we examine the decrease in intensity levels with distance at three different altitudes, 10 km, 50 km, and 100 km.

If we move ~3 m away from the source, the intensity levels drop to ~85 dB, ~65 dB, and ~39 dB for, respectively. At ~10 m away, these intensities drop to ~74 dB, ~54 dB, and ~28 dB, respectively. At ~50 m away, these intensities drop to ~60 dB, ~40 dB, and ~14 dB, respectively. And at ~150 m away, the intensities drop to ~51 dB, ~31 dB, and ~5 dB, respectively. For comparison at sea level, the intensities would be ~90 dB, ~66 dB, and ~56 dB at distances of ~3 m, ~50 m, and ~150 m, respectively.

Thus, at 100 km altitude one need only move a little over 100 m away from the source before the intensity level drops below the hearing threshold (i.e., ~5 dB for a 20 year old male at 1000 Hz).

The model only went to 100 km but even so, our source would become difficult to hear if we moved a little more than ~100 m from it. Given that the density decreases exponentially with an e-folding distance of only ~8.5 km (pressure does so similarly as well), if we extrapolate our estimates for $L_{i,src}\left( h \right)$ then the value drops to ~10 dB by ~177 km.

So by ~200 km a human probably could not hear a source ~1 m away that produced a 100 dB, 1000 Hz intensity level at sea level.