A small bead of mass is threaded onto an otherwise uniform string of characteristic…

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

A small bead of mass

m

is threaded onto an otherwise uniform string of characteristic impedance

Z

; the bead presents a load impedance

Z_{\rm L}=Z+i\omega m

to a wave arriving from one side, so that the power transmission is

|t|^2 = \left[1+(\omega m/2Z)^2\right]^{-1}

. Such a bead is found to transmit

80\%

of the incident power at a frequency of

50\,\mathrm{Hz}

. ((a)) Find the crossover frequency

f_{\rm c}

, defined as the frequency at which the transmitted and reflected powers are equal. ((b)) Find the fraction of the power transmitted at

150\,\mathrm{Hz}

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