A chain has sound speed and spacing

PHYSICS · Subjective · Fermi University Physics · Oscillations, Waves & Acoustics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids

Condensed Matter Physics, Ch Elastic waves and the vibrating chain

Problem

A chain has sound speed vs=5000 m s−1v_s=5000\,\mathrm{m\,s^{-1}} and spacing a=2.80 A˚a=2.80\,\mathrm{\mathring{A}}. One argument runs like this: “At q=0.9 π/aq=0.9\,\pi/a, close to the zone boundary, the phase velocity ω/q\omega/q is still nearly 3500 m s−13500\,\mathrm{m\,s^{-1}}, comparable to the sound speed itself, so a disturbance must still be moving through the chain nearly that fast.” Compute the group velocity vgv_g at this qq, and use it to say what is wrong with this reasoning.

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