# Experimental Values for Amplitude $\delta^*$ of Particle Displacement in Acoustic Fields?

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.

• Jan 29, 2022 at 20:47

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 $$p^*$$.