A string with non-dispersive speed parameter rests on an elastic bed of cutoff…

PHYSICS · Subjective · Fermi University Physics · Oscillations, Waves & Acoustics · Superposition, boundaries and interference · Superposition, boundaries and interference

Modes, Packets and Media, Ch Reflection, transmission, matching and tunnelling at boundaries

Problem

A string with non-dispersive speed parameter v=8.0 m s−1v = 8.0\,\mathrm{m\,s^{-1}} rests on an elastic bed of cutoff frequency ωc=15 rad s−1\omega_{\rm c}=15\,\mathrm{rad\,s^{-1}} (so ω2=ωc2+v2k2\omega^2 = \omega_{\rm c}^2+v^2k^2 inside the bed) over a stretch of length d=1.0 md=1.0\,\mathrm{m}; the string is bare (no bed) elsewhere. A wave of angular frequency ω=10 rad s−1\omega = 10\,\mathrm{rad\,s^{-1}} (ω<ωc\omega<\omega_{\rm c}) is incident on the barrier from the bare string. Use T=[1+(k2+α2)2sinh⁡2αd4k2α2]−1T = \left[1+\frac{(k^2+\alpha^2)^2\sinh^2\alpha d}{4k^2\alpha^2}\right]^{-1} where k=ω/vk=\omega/v is the wavenumber on the bare string and α=ωc2−ω2/v\alpha=\sqrt{\omega_{\rm c}^2-\omega^2}/v is the decay constant inside the bed. Find the power transmission coefficient TT.

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