7.7 Negative energy waves 227Since odd powers of oscillating quantities vanish upon time averaging, Eq.(7.96) becomes
w ̄sys =
ε 0 |E ̃|^2
4+n 0 me〈
1
2
v^21 +
n 1
n 0v 1 v 0〉
=
ε 0 |E ̃|^2
4+
1
2
n 0 me〈
v^21〉
+mev 0 〈n 1 v 1 〉. (7.98)The linearized continuity equation gives
−i(ω−kv 0 ) ̃n+n 0 ik ̃v=0 (7.99)or
̃n
n 0
=
k ̃v
ω−kv 0. (7.100)
Thefluid quiver velocity in the wave is
v ̃=qE ̃
−i(ω−kv 0 )me(7.101)
so that
〈
v^21
〉
=
1
2
q^2 E^2
(ω−kv 0 )^2 m^2 e(7.102)
and
〈n 1 v 1 〉=
k
2n 0 q^2 E^2
(ω−kv 0 )^3 m^2 e. (7.103)
We may now evaluate Eq.(7.98) to obtain
w ̄sys =
ε 0 |E ̃|^2
4+n 0 me[
q^2 E^2
4(ω−kv 0 )^2 m^2 e+
kv 0
2q^2 E^2
(ω−kv 0 )^3 m^2 e]
=
ε 0 |E ̃|^2
4[
1+
ω^2 pe
(ω−kv 0 )^2+
2 ω^2 pekv 0
(ω−kv 0 )^3]
=
ε 0 |E ̃|^2
2[
1 ±
kv 0
ωpe]
(7.104)
where Eq.(7.92) has been invoked repeatedly. This is the same as Eq. (7.93). The energy
flux associated with this wave shows is also negative (cf. assignments). However, the group
velocity is positive (cf. assignments) because the group velocity is the ratio of a negative
energyflux to a negative energy density.
Dissipation acts on negative energy waves in a manner opposite to the way it acts
on positive energy waves. This can be seen by Taylor-expanding the dispersion relation
P(ω,k)=0as done in Eq.(5.83)
ωi=−Pi
[∂Pr/∂ω]ω=ωr. (7.105)
Expanding Eq.(7.91) gives
w ̄=ε 0 ω|E|^2
4∂P
∂ω