Fundamentals of Plasma Physics

14.3 Echoes 425

Theφ ̃

lower

(t,x)term is obtained by lettingka−kb→−(ka−kb)and so

̃φ 2 (t,x)= ̃φupper 2 (t,x)+ ̃φlower(t,x). (14.147)

If it is assumed thatf ̄ 0 =(πvT)−^1 /^2 exp(−v^2 /v^2 T)then the velocity integrals in the upper
and lower terms will be

1

√

π

∫

dve−v

(^2) /vT (^2) ±i[(kb−ka)t−kbτ]v
= exp

(

−[(kb−ka)t−kbτ]^2 v^2 T/ 4

)

= exp

(

−

[

t−

kb

kb−ka

τ

] 2

(kb−ka)^2 vT^2

4

)

.

(14.148)

Phase mixing due to the extreme velocity dependence of theexp(i(kb−ka)vt−ikbvτ)
factor will cause the velocity integral to vanish except at the special time

techo=

kb

kb−ka

τ (14.149)

when there is no phase mixing and so a finite ̃φ 2 (t,x)signal results. This is the echo. The
half-width of the echo can be determined by writing Eq.(14.148) as

1

√

π

∫

dve−v

(^2) /vT (^2) ±i[(kb−ka)t−kbτ]v
=e−(t−techo)
(^2) /(∆t) 2
(14.150)
where
∆t=

2

|kb−ka|vT

(14.151)

is the width of the echo.

14.3.5Spatial echoes

Creating spatially periodic sources with temporal delta functions is experimentally more
difficult than creating temporally periodic sources with spatial delta functions. In the latter
arrangement, two spatially separated grids are placed in a plasma and each grid is excited at
a different frequency. This system is then characterized by a Fourier transform in time and
a Laplace transform in space so that the convective derivative in the linearized Vlasov equa-
tion has the form−iωf 1 +v∂f 1 ∂xgiving ballistic terms proportional toexp(iωx/v)instead
of proportional toexp(−ikvt).Thus, if two grids separated by a distanceLare excited at
respective frequenciesω 1 andω 2 , one grid will excite a ballistic term∼exp(iω 1 (x−L)/v)
and the other will excite a ballistic term∼exp(iω 2 x/v)and so the non-linear product of
these two ballistic terms will include a factor∼exp(iω 1 (x−L)/v−iω 2 x/v)which will
phase mix to zero except at the spatial location

xecho=

ω 1 L

ω 1 −ω 2

. (14.152)

If the spatial Landau damping length of the two linear modes is much less thanxecho,
then the linear modes will appear to damp away spatially and then the echo will appear
atx=xechowhich will be much further away. This sort of arrangement was used to
demonstrate echoes in lab experiments by Malmberg et al. (1968)