Two strings of impedances and are to be joined through a taper along which rises…

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 strings of impedances

Z_1

and

Z_2=4Z_1

are to be joined through a taper along which

\ln Z

rises linearly over a length

\ell

. Tension and density are varied together so that the wave speed stays

v=20\,\mathrm{m\,s^{-1}}

throughout, and the taper formula of the text may be used as it stands. The join must reflect less than

1

percent of the incident power at every frequency from

f_{\rm min}=50\,\mathrm{Hz}

upwards. ((a)) Find the largest value of

|\mathrm{sinc}(k\ell)|

this allows. ((b)) Find the shortest taper length

\ell

that meets the requirement.

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