Two identical stretches of elastic bed

PHYSICS · Multi-part · 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

Two identical stretches of elastic bed, each of cutoff frequency

\omega_{\rm c}=20\,\mathrm{rad\,s^{-1}}

, lie under a string of speed parameter

v=8.0\,\mathrm{m\,s^{-1}}

, a gap

L

of bare string apart. The string is driven at

\omega=\omega_{\rm c}/\sqrt2

, and each bed is exactly

\pi

decay lengths thick,

\alpha d=\pi

. Seen from the gap, each bed reflects with

r=i\tanh\alpha d

. ((a)) Find the wavenumber

k

on the bare string, the decay constant

\alpha

in the beds, and each bed's thickness

d

. ((b)) From the magnitude and phase of

r

, state the condition on

kL

for the pair to transmit completely, and find the two smallest gaps

L

that satisfy it.

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