Fundamentals of Plasma Physics

126 Chapter 4. Elementary plasma waves

so Eq. (4.12) becomes

mσ

∂^2 nσ 1

∂t^2

=qσnσ 0 ∇^2 φ 1 +γκTσ 0 ∇^2 nσ 1 (4.16)

wherePσ 0 =nσ 0 κTσ 0 has been used.
Although this system of linear equations could be solved by the formal method of
Fourier transforms, we instead take the shortcut of making the simplifying assumption
that the linear perturbation happens to be a single Fourier mode. Thus, it is assumed that
alllinearized dependent variables have the wave-like dependence

nσ 1 ∼exp(ik·x−iωt), φ 1 ∼exp(ik·x−iωt), etc. (4.17)

so that∇→ikand∂/∂t →−iω.Equation (4.16) therefore reduces to the algebraic
equation
mσω^2 nσ 1 =qσnσ 0 k^2 φ 1 +γκTσ 0 k^2 nσ 1 (4.18)
which may be solved fornσ 1 to give

nσ 1 =

qσnσ 0

mσ

k^2 φ 1

(ω^2 −γk^2 κTσ 0 /mσ)

. (4.19)

Poisson’s equation provides another relation betweenφ 1 andnσ 1 , namely

−k^2 φ 1 =

1

ǫ 0

∑

σ

nσ 1 qσ. (4.20)

Equation (4.19) is substituted into Poisson’s equation to give

k^2 φ 1 =

∑

σ

nσ 0 q^2 σ

ǫ 0 mσ

k^2 φ 1

(ω^2 −γk^2 κTσ 0 /mσ)

(4.21)

which may be re-arranged as
[
1 −

∑

σ

ω^2 pσ

(ω^2 −γk^2 κTσ 0 /mσ)

]

φ 1 = 0 (4.22)

where

ω^2 pσ≡

nσ 0 q^2 σ

ε 0 mσ

(4.23)

is the square of theplasma frequencyof speciesσ.A useful way to recast Eq.(4.22) is

(1 +χe+χi)φ 1 = 0 (4.24)

where

χσ=−

ω^2 pσ

(ω^2 −γk^2 κTσ 0 /mσ)

(4.25)

is called the susceptibility of speciesσ.In Eq.(4.24) the “1” comes from the “vacuum” part
of Poisson’s equation (i.e., the LHS term∇^2 φ)while the susceptibilities give the respective
contributions of each species to the right hand side of Poisson’s equation. This formalism
follows that of dielectrics where the displacement vector isD=εEand the dielectric con-
stant isε= 1 +χwhereχis a susceptibility.