A one-dimensional chain of identical masses linked by identical springs has the…

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

Modes, Packets and Media, Ch Dispersion and the propagation of wave packets

Problem

A one-dimensional chain of identical masses linked by identical springs has the dispersion relation ω(k)=ωmax⁡sin⁡(ka/2)\omega(k) = \omega_{\max}\sin(ka/2) for 0≤ka≤π0 \le ka \le \pi, where ωmax⁡=4000 rad s−1\omega_{\max} = 4000\,\mathrm{rad\,s^{-1}} is the top of the band and a=1.0 mma = 1.0\,\mathrm{mm} is the lattice spacing. Calculate the phase velocity vpv_{\rm p} and the group velocity vgv_{\rm g} at ka=3.0 radka = 3.0\,\mathrm{rad}, close to the band edge at ka=π≈3.14ka = \pi \approx 3.14.

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