Fundamentals of Plasma Physics

14.3 Echoes 423

from thebpulse in the second ̃φextfactor in Eq.(14.135) will be considered. We therefore
substitute theacontribution in Eq.(14.137) for the first ̃φextfactor in Eq.(14.135) to obtain

̃φ 2 (p,k) = ωpπφa

k^2 D(p,k)

e

m

∫

dv

∫∞

−∞

dk′

2 π

∫b+i∞

b−i∞

dp′

2 π

ikk′δ(k′±ka)

(p+ikv)^2 D(p′,k′)

×

i ̄k′

(p−p′)+i ̄k′v

̃φext((p−p′), ̄k′)

D(p−p′, ̄k′)

∂f ̄ 0

∂v

. (14.138)

The effect of theδ(k′±ka)factor when evaluating thek′integral is to forcek′→±ka
and also ̄k′→k∓kaso that

φ ̃ 2 (p,k) = ωp

k^2 D(p,k)

eφa

2 m

∑

±

∫

dv

∫b+i∞

b−i∞

dp′

2 π

(∓ikka)

(p+ikv)^2 D(p′,∓ka)

×

i (k∓ka)

p ̄′+i(k ∓ka)v

φ ̃ext( ̄p′,(k∓ka))

D( ̄p′,k∓ka)

∂f ̄ 0

∂v

. (14.139)

Now let us substitute for the secondφ ̃extfactor using the contribution from thebpulse to
obtain

φ ̃ 2 (p,k) = π

2 k^2 D(p,k)

e

m

φaφb

∑

±

∑

±

∫

dv

∫b+i∞

b−i∞

dp′

2 π

(∓ikka)

(p+ikv)^2 D(p′,∓ka)

×

i (k∓ka)

p ̄′+i(k ∓ka)v

δ(k∓ka±kb)e−p ̄

′τ

D( ̄p′,k∓ka)

∂f ̄ 0

∂v

. (14.140)

The upper choice of the±and∓signs is selected and the inverse Fourier transform per-
formed to obtain

̃φupper 2 (p,x) = φaφb

4 k^2 D(p,k)

e

m

∫∞

−∞

dk

∫

dv

∫b+i∞

b−i∞

dp′

2 π

(−ikka)

(p+ikv)^2 D(p′,−ka)

×

i (k−ka)

p ̄′+i(k−ka)v

δ(k−ka+kb)e− ̄p

′τ+ikx

D( ̄p′,k−ka)

∂f ̄ 0

∂v

=

φaφb

4(ka−kb)^2 D(p,ka−kb)

e

m

∫

dv

∫b+i∞

b−i∞

dp′

2 π

×

i(ka−kb)

(p+i(ka−kb)v)^2

ka

D(p′,−ka)

ikb

p ̄′−ikbv

e− ̄p

′τ+i(ka−kb)x

D( ̄p′,−kb)

∂f ̄ 0

∂v

.

(14.141)

The lower choice of the±and∓signs means thatka→−kaandkb→−kband so at

the end of the calculationφ ̃

lower

2 (p,x)can also be determined by simply lettingka→−ka