[Physics] How far do air particles move when a sound wave passes through them


How far do air particles move when a sound wave passes through them? I know that they don't actually travel, the question is how far do they oscillate or what is the physical amplitude of the oscillation?

Obviously the answer is different for different mediums and volumes (and maybe frequency?), so for the purposes of the question, assume we're talking about a 100Hz sine wave, at about 75db (low bass, at listening volume), travelling through air at sea-level pressure.

Best Answer

Sound pressure level (SPL) in dB is defined relative to a pressure $p_{ref}=20\mu Pa$ $$L_p=20\log_{10}\left(\frac{p_{rms}}{p_{ref}}\right )$$ 75dB corresponds to acoustic pressure of 0.11 Pa, you can use this online calculator to easily check other SPLs.

Acoustic velocity is proportional to acoustic pressure through acoustic impedance $Z=\rho c$ where $\rho$ is air density and $c$ is sound velocity, for air at room temperature $Z \approx 400 \frac{Ns}{m^3}$.

With acoustic pressure and impedance you can calculate all sorts of quantities. In particular, particle displacement is calculated as $$ \xi =\frac{p}{Z\omega}=\frac{p}{Z2\pi f} $$ where $f$ is the acoustic frequency. Plugging all numbers into this formula, we get $\xi=4.4 \cdot 10^{-7}\,m$. Keep in mind that SPL is given in logarithmic scale, so if you take $L_p=150\,dB$ then $\xi= 0.0025\,m=2.5\,mm$.