A radio wave travelling through a region of ionised gas (an ionospheric plasma) obeys…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Optics & Photonics · Wave propagation and light in media

Optics and Photonics, Ch Harmonic waves, phasors and group velocity

Problem

A radio wave travelling through a region of ionised gas (an ionospheric plasma) obeys the dispersion relation

\omega^2 = \omega_p^2 + c^2k^2

where

\omega_p

is the (constant) plasma frequency of the region and

c

is the speed of light in vacuum,

c = 3.00\times10^8\ \mathrm{m\,s^{-1}}

. Take

\omega_p = 4.00\times10^8\ \mathrm{rad\,s^{-1}}

and consider a wave packet built from wavenumbers close to

k = 1.00\ \mathrm{rad\,m^{-1}}

(a propagating mode, since

\omega > \omega_p

here). ((a)) Before calculating anything, state whether you expect the packet's envelope to travel faster or slower than its individual wave crests, and justify your answer in one sentence from the shape of

\omega(k)

alone. ((b)) Calculate the phase velocity

v = \omega/k

and the group velocity

v_g = d\omega/dk

at this

k

, and check that your numbers agree with your prediction.

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