A WR-90 waveguide has inner width and carries its lowest mode at

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Electricity, Magnetism & Electromagnetism · Waves in two and three dimensions, dispersion and wave packets

Modes, Packets and Media, Ch Transmission lines and electromagnetic waves in matter, plasmas and guides

Problem

A WR-90 waveguide has inner width

a = 22.86\,\mathrm{mm}

and carries its lowest mode at

f = 10.0\,\mathrm{GHz}

. Someone argues: "The cutoff frequency is

f_{\rm c} = c/2a = 6.56\,\mathrm{GHz}

. The wave inside the guide is two plane waves crossing at an angle

\theta

to the axis, with

\cos\theta = \sqrt{1 - (f_{\rm c}/f)^2} = 0.755

. The phase velocity is

v_{\rm p} = c/\cos\theta = 1.325c = 3.97\times10^{8}\,\mathrm{m\,s^{-1}}

. Since this is the speed at which the wave travels down the guide, a pulse crossing

L = 3.00\,\mathrm{m}

of this guide takes

t = L/v_{\rm p} = 3.00/(3.97\times10^{8}) = 7.56\,\mathrm{ns}

." ((a)) Identify the error in this reasoning. ((b)) Give the correct transit time for the pulse over

3.00\,\mathrm{m}

of guide at

10.0\,\mathrm{GHz}

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