Fundamentals of Plasma Physics

202 Chapter 6. Cold plasma waves in a magnetized plasma

whereE ̃(k)is the amplitude of the mode with wavenumberk.The dispersion relation
assigns anωto eachk, so that at later times the field evolves as

E(x,t)=

∫

dkE ̃(k)exp(ik·x−iω(k)t) (6.88)

whereω(k)is given by the dispersion relation. The integration overkmay be viewed as a
summation of rapidly oscillatory waves, each having different rates of phase variation. In
general, this sum vanishes because the waves add destructively or “phase mix”. However,
if the waves add constructively, a finiteE(x,t)will result. Denoting the phase by

φ(k)=k·x−ω(k)t (6.89)

it is seen that the Fourier components add constructively at extrema (minima or maxima)
ofφ(k),because in the vicinity of an extrema, the phase is stationary with respect tok,
that is, the phase does not vary withk. Thus, the trajectoryx=x(t)along whichE(x,t)
is finite is the trajectory along which the phase is stationary. At timetthe stationary phase
is the place where∂φ(k)/∂kvanishes which is where

∂φ

∂k

=x−

∂ω

∂k

t=0. (6.90)

The trajectory of the points of stationary phase is therefore

x(t)=vgt (6.91)

wherevg=∂ω/∂kis called the group velocity. The group velocity is the velocity at
which a pulse propagates in a dispersive medium and is also the velocity at which energy
propagates.
The phase velocity for a one-dimensional system is defined asvph=ω/k.In three
dimensions this definition can be extended to bevph=ˆkω/k, i.e. a vector in the direction
ofkbut with the magnitudeω/k.
Group and phase velocities are the same only for the special case whereωis linearly
proportional tok,a situation which occurs only if there is no plasma. For example, the
phase velocity of electromagnetic plasma waves (dispersionω^2 =ω^2 pe+k^2 c^2 ) is

vph=ˆk

√

ω^2 pe+k^2 c^2

k

(6.92)

which is faster than the speed of light. However, no paradox results because information
and energy travel at the group velocity, not the phase velocity. The group velocity for this
wave is evaluated by taking the derivative of the dispersion with respect tokgiving

2 ω

∂ω

∂k

=2kc (6.93)

or
∂ω
∂k

=

kc

√

ω^2 pe+k^2 c^2

(6.94)

which is less than the speed of light.
This illustrates an important property of the wave normal surface concept – awave
normal surface is a polar plot of thephase velocityand should not be confused with the
group velocity.