226 Chapter 7. Waves in inhomogeneous plasmas and wave energy relations
total system energy density in going from a situation where there is no wave to a situation
where there is a wave. Typically, negative energy waves occur when the equilibrium has a
steady-stateflow velocity and there exists a mode which causes the particles to develop a
slower mean velocity than in steady state. Wave growth taps free energy from theflow.
As an example of a negative energy wave, we consider the situation where unmagne-
tized cold electrons stream with velocityv 0 through a background of infinitely massive
ions. As shown earlier, the electrostatic dispersion for this simple1D situation withflow
involves a parallel dielectric involving a Doppler shifted frequency,i.e., the dispersion re-
lation is
P(ω,k)=1−ω^2 pe
(ω−kv 0 )^2=0. (7.90)
Since the plasma is unmagnetized, its dielectric tensor is simply
←→
K(ω,k)=P(ω,k)←→
I.
Using Eq.(7.79), the wave energy density is
w=
ε 0 |E|^2
4∂
∂ω(ωP(ω,k))=
ε 0 |E|^2
2ωω^2 pe
(ω−kv 0 )^3. (7.91)
However, the dispersion relation, Eq.(7.90), shows that
ω=kv 0 ±ωpe (7.92)so that Eq.(7.91) can be recast as
w=
ε 0 |E|^2
2(
1 ±
kv 0
ωpe)
. (7.93)
Thus, ifkv 0 >ωpeand the minus sign is selected, the wave hasnegativeenergy density.
This result can be verified by direct calculation of the change in system energy density
due to growth of the wave. When there is no wave, the electric field iszero and the system
energy densitywsysis simply the beam kinetic energy density
wsys 0 =1
2
n 0 mev 02. (7.94)Now consider a one dimensional electrostatic wave with electric fieldE=Re
(
E ̃eikx−iωt)
.
The system average energy density with this wave is
wsyswave=
ε 0 |E ̃|^2
4+
〈
1
2
n(x,t)mev(x,t)^2〉
(7.95)
so that the change in system energy density due to the wave is
w ̄sys=wwavesys −w 0 sys=
ε 0 |E ̃|^2
4+
〈
1
2
[n 0 +n 1 (x,t)]me[v 0 +v 1 (x,t)]^2〉
−
1
2
n 0 mev^20
(7.96)
where
n 1 (x,t)=Re
(
n ̃eikx−iωt)
, v 1 (x,t)=Re(
̃veikx−iωt