Fundamentals of Plasma Physics

6.6 Group velocity 201

whistler dispersion for acoustic (a few kHz) waves in the ionosphere is

n^2 −=

ω^2 pe

∣

∣

∣

ωce

ω

cosθ

∣

∣

∣

(6.81)

or

k=

ωpe

c

√

ω

|ωcecosθ|

. (6.82)

Each frequencyωin Eq.(6.80) has a correspondingkgiven by Eq.(6.82) so that the distur-
banceg(x,t)excited by a lightning bolt has the form

g(x,t)=

1

2 π

∫

eik(ω)x−iωtdω (6.83)

wherex=0is the location of the lightning bolt. Because of the strong dependence ofk
onω,contributions to the phase integral in Eq.(6.83) at adjacent frequencies will in general
have substantially different phases. The integral can then be considered as the sum of
contributions having all possible phases. Since there will be approximately equal amounts
of positive and negative contributions, the contributions will cancel each other out when
summed;this cancelling is called phase mixing.
Suppose there exists some frequencyωat which the phasek(ω)x−ωthas a local
maximum or minimum with respect to variation ofω.In the vicinity of this extremum,
the phase is independent of frequency and so the contributions from adjacent frequencies
constructively interfere and produce a finite signal. Thus, an observer located at some
positionx = 0will hear a signal only at the time when the phase in Eq.(6.83) is at an
extrema. The phase extrema is found by setting to zero the derivative of thephase with
respect to frequency, i.e. setting
∂k
∂ω

x−t=0. (6.84)

From Eq.(6.82) it is seen that

∂k

∂ω

=

ωpe

2 c

1

√

ω|ωcecosθ|

(6.85)

so that the time at which a frequencyωis heard by an observer at locationxis

t=

ωpe

2 c

x

√

ω|ωcecosθ|

. (6.86)

This shows that lower frequencies are heard at later times, resultingin the descending tone
characteristic of whistlers.

6.6 Group velocity

Suppose that at timet=0the electric field of a particular fast or slow mode is decomposed
into spatial Fourier modes, each varying asexp(ik·x). The total wave field can then be
written as

E(x)=

∫

dkE ̃(k)exp(ik·x) (6.87)