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I've been searching but have been unable to find any experimental values for the amplitude $\delta^*$ of the particle displacement from equilibrium that particles in a medium such as air will undergo when a sound wave propagates through that medium. This parameter $\delta^*$ shows up a lot in theoretical treatments of sound, such as here, and sure we can relate it to a bunch of other parameters like amplitude $v^*$ of particle velocity and the temporal angular frequency $\omega$ of the sound wave, but ultimately I would be interested to know if there are any experimentally tabulated values for $\delta^*$ in various different circumstances.

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You probably won't find it tabulated anywhere because particle displacement ($\delta^*$) and likewise particle velocity ($v^*$) and sound pressure ($p^*$) all depend on the loudness of the sound.

Loudness is often measured in a logarithmic dB (dezibel) scale. The sound pressure level $L_p$ is defined as $$L_p=20 \log_{10}\left(\frac{p^*}{p^*_\text{ref}}\right) \text{dB}$$ or equivalently $$p^*=p^*_\text{ref} 10^{L_p/20\text{ dB}}$$ where is $p^*$ is the sound pressure and $p^*_\text{ref}$ is a reference sound pressure (for air it is $p^*_\text{ref}=2\cdot 10^{-5}$ Pa). For example:

  • $L_p=0$ dB (corresponding to $p^*=2\cdot 10^{-5}$ Pa) is so quiet that it is hardly perceivable.
  • $L_p=130$ dB (corresponding to $p^*=63$ Pa) is so loud that it damages the ear.

With the relations $v^*=\omega\delta^*$ and $p^*=\rho c\omega\delta^*$ you can then calculate particle displacement $\delta^*$ and particle velocity $v^*$ from the sound pressure $p^*$.

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  • $\begingroup$ Good answer with concentrated information. I just wanna make a minor correction here. The term "loudness" is used for the subjective feeling of how "strong" a sound is, which has a weaker connection to pressure (and displacement) amplitude than the sound pressure level (it is also frequency dependent). Additionally, level refers to the logarithmic (in dB) counterpart of a quantity in acoustics. For example Sound Pressure Level is actually pressure converted to a logarithmic (dB) scale, with some reference value always in effect. +1 from me. $\endgroup$
    – ZaellixA
    Feb 1, 2022 at 0:20

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