15.3 Application to waves 437
frequency modes, then the ponderomotive force contains a term at the difference (i.e.,
beat) frequency between the frequencies of the two modes. Thus, at low frequencies
the electron equation of motion, Eq.(15.34) becomes
∂ ̃ue
∂t
=
qe
me
E ̃−^1
men
∇P ̃e−
1
2
∇
(
̃uhe
) 2
(15.39)
where only the beat frequency component in
(
u ̃he
) 2
is used. The ponderomotive force
provides a mechanism for high-frequency waves to couple to low frequency waves.
It acts as an effective pressure scaling asmen
(
u ̃he
) 2
/ 2 and so in a sense, the quiver
velocityu ̃heacts as a thermal velocity. The ratio of ion radiation pressure to the ion
pressure is smaller than the corresponding ratio for electrons by a factorofme/mi
because the ion quiver velocity is smaller by this factor. Thus, ion ponderomotive
force is ignored since it is so small.
- Beating of a low frequency wave with a high frequency wave to drive another high
frequency wave (modulation).By writing Ampere’s law as
∇×B ̃=μ 0
∑
σ
( ̃nσqσ ̃uσ+n 0 qσ ̃uσ)+μ 0 ε 0
∂E ̃
∂t
(15.40)
it is seen that densityfluctuations provide a nonlinear component to the current den-
sity. The nonlinear term can be put on the right hand side to emphasize its role as a
nonlinear driving term so that Ampere’s law becomes
∇×B ̃−μ 0 ε 0
∂ ̃E
∂t
−μ 0
∑
σ
n 0 qσ ̃uσ=−μ 0
∑
σ
̃nσqσ ̃uσ. (15.41)
The nonlinear term is assumed to be a product of a high frequency wave and a low
frequency wave. The linearized continuity equation gives
∂n ̃σ
∂t
=−nσ∇· ̃uσ (15.42)
showing thatn ̃σ∼ ̃uσ/ωso that the productnlσuhσis much larger than the product
nhσulσwherelandhrefer to low and high frequency waves. Thus, the dominant effect
of a low frequency wave is to modulate the density profile seen by a high frequency
wave.
15.3.2Possible types of wave interaction
As discussed in Sec.4.2, three distinct types of waves can propagate in anunmagne-
tized uniform plasma and these waves have the dispersion relations:
ω^2 = ω^2 pe+k^2 c^2 ,electromagnetic wave
ω^2 = ω^2 pe
(
1+3k^2 λ^2 de
)
,electron plasma wave
ω^2 =
k^2 c^2 s
1+k^2 λ^2 De
,ion acoustic wave. (15.43)