On a string on an elastic bed with and cutoff frequency

PHYSICS · Subjective · Fermi University Physics · Oscillations, Waves & Acoustics · Waves in two and three dimensions, dispersion and wave packets · Waves in two and three dimensions, dispersion and wave packets

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

Problem

On a string on an elastic bed with v=5.0 m s−1v = 5.0\,\mathrm{m\,s^{-1}} and cutoff frequency ωc=10 rad s−1\omega_{\rm c} = 10\,\mathrm{rad\,s^{-1}}, a wave packet is launched with carrier frequency ω=3ωc\omega = 3\omega_{\rm c} and an initially unchirped Gaussian envelope of width σ0=0.80 m\sigma_0 = 0.80\,\mathrm{m}. Using ω′′=v2ωc2/ω3\omega'' = v^2\omega_{\rm c}^2/\omega^3 and the width law σ(t)=σ02+(ω′′t/2σ0)2\sigma(t) = \sqrt{\sigma_0^2 + (\omega''t/2\sigma_0)^2}, find the envelope width after t=50 st = 50\,\mathrm{s}.

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