Bending (flexural) waves on a long

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Optics & Photonics · Wave propagation and light in media

Optics and Photonics, Ch Harmonic waves, phasors and group velocity

Problem

Bending (flexural) waves on a long, thin, stiff steel beam obey the dispersion relation ω(k)=αk2\omega(k) = \alpha k^2 with α=1.20×10−3 m2 s−1\alpha = 1.20\times10^{-3}\ \mathrm{m^2\,s^{-1}}, a constant fixed by the beam's stiffness and mass per unit length. Consider a wave packet built from wavenumbers close to k=50.0 rad m−1k = 50.0\ \mathrm{rad\,m^{-1}}. ((a)) Before calculating anything, state whether you expect the packet's envelope to run ahead of or behind its individual crests, arguing from whether ω\omega grows faster or slower than in direct proportion to kk. ((b)) Calculate the phase velocity v=ω/kv=\omega/k and the group velocity vg=dω/dkv_g=d\omega/dk at this kk, and confirm your prediction.

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