A string on an elastic bed has with and .

PHYSICS · Multi-part · 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

A string on an elastic bed has

\omega^2=\omega_{\rm c}^2+v^2k^2

with

v=12\,\mathrm{m\,s^{-1}}

and

\omega_{\rm c}=25\,\mathrm{rad\,s^{-1}}

. ((a)) Drive the string at

\omega=37.5\,\mathrm{rad\,s^{-1}}

(above cutoff). Find the wavenumber

k

, the phase velocity

v_{\rm p}

, the group velocity

v_{\rm g}

, and (quoting that on this lossless string the energy velocity equals the group velocity) the energy velocity

v_{\rm E}

. ((b)) Drive the string at

\omega=12.5\,\mathrm{rad\,s^{-1}}

(below cutoff). Show that no travelling wave exists at this frequency and find the spatial decay constant

\alpha

of the resulting evanescent disturbance. ((c)) State the front velocity of the string in both cases and explain, in one or two sentences, why it does not depend on whether the drive frequency is above or below cutoff, and which of the five velocities in this problem is the one that limits how fast a signal can be sent.

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