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The name itself suggests that we can hear acoustic branch of phonon modes (and not the optical branch), but other than that I do not see any reason whatsoever why we are able to hear them.

In order to be completely clear, I define the acoustic phonons as the ones with the property that $\omega \propto k$ when $k \to 0$. (while the optical phonons are the ones that do not have that property).

Why are our ears able to detect a disturbance with $\omega \propto k$ property?

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This is a phonon dispersion plot for silicon carbide:

enter image description here

The $\Gamma$ point refers to $k=0$; the designations of the branches are transverse/longitudinal and acoustic/optical. $X,K,$ and $L$ refer to different points in the 3D Brillouin zone, and correspond to special high-symmetry values of the wavevector $\vec k$.

Noting that the upper limit for human hearing is on the order of 20 kHz, it should be immediately clear why the optical branches are inaccessible to our ears, as are the vast majority of the acoustic branches; we are able to hear only the very lowest frequencies, which exist as the $\vec k\rightarrow 0$ limit of the acoustic branches.

On the other hand, the frequencies of the optical branches correspond to the infrared range of electromagnetic radiation, and can be excited by (infrared) light pulses.

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