What is the displacement amplitude of a 140 db sound wave? I am teaching a unit on sound in my upper elementary science class, and my students are curious about how far air molecules are actually being displaced in a sound wave. I told them that if the air molecules were actually oscillating a foot or more they would probably be killed by the pressure. I would like to get actual, accurate information about this. I'm not a sound expert. I do know that the 'threshold of pain' for sounds is around 140 dB, so I am assuming that would be a good place to start in terms of measuring the amplitude of a sound wave.
So how far the closest answer I've been able to find is that

"A sound with an intensity of 1*10-12 W/m2 corresponds to a sound that will displace particles of air by a mere one-billionth of a centimeter. The human ear can detect such a sound."
(https://www.physicsclassroom.com/class/sound/Lesson-2/Intensity-and-the-Decibel-Scale)

If I were to measure a sound wave's amplitude in meters, are they measured in nanometers? How high of amplitude would be dangerous to a human?
 A: There are two equations you need. Firstly the sound intensity in decibels (the SPL) is related to the RMS sound pressure by:
$$ SPL = 20 \log_{10}\left(\frac{\Delta P_{rms}}{2 \times 10^{-5}} \right) $$
We are going to need the pressure so we'll rearrange this to:
$$ \Delta P_{rms} = 2 \times 10^{-5} \times 10^{SPL/20} \tag{1} $$
Now the second equation. This relates the pressure $\Delta P_m$ to the displacement $\Delta s_m$ by:
$$ \Delta P = v\rho\omega \Delta s \tag{2} $$
So we can combine equations (1) and (2) to give the rather unwieldly equation:
$$ 2 \times 10^{-5} \times 10^{SPL/20} = v\rho\omega \Delta s $$
And rearranging gives us the equation we need for the maximum displacement of the air molecules in a sound wave.
$$ s = \frac{2 \times 10^{-5} \times 10^{SPL/20}}{v\rho\omega} \tag{3} $$
In this equation $v$ is the speed of sound ($343$ m/s), $\rho$ is the density of the air ($1.2$ kg/m³) and $\omega$ is the angular frequency i.e. $2\pi$ times the frequency in hertz. And since we used the RMS pressure equation (3) will give us the RMS displacement.
So for example if we take the SPL to be $140$ dB and the frequency to be $1$ kHz we get the displacement to be 78 microns.
I wish I had a reference for all this, but I found the equations in some old notes with the comment "this is proved in the textbook". Sadly I didn't write down which textbook.
