15.3 Application to waves 441However, the high frequency linear electron equation of motion is
∂ ̃ue
∂t=
qe
meE ̃−^3 κTe
men∇ ̃ne (15.62)so Eq.(15.61) becomes
∂^2 ̃ne
∂t^2+ ω^2 pen ̃e−3 κTe
me∇^2 n ̃e=−∂
∂t∇·( ̃ne ̃ue). (15.63)In principle, there are two possible components in the right hand side nonlinear term since
̃ne ̃ue= ̃nle ̃uhe+ ̃nhe ̃ulewherelandhrefer to low and high frequency waves. If both high
and low frequency waves are electrostatic, then ̃ue∼E ̃=−ikφ ̃and∇· ̃ue=ik· ̃ue=ik·
̃ue∼k^2 ̃φwhereφ ̃is the electrostatic potential. The linearized continuity equation gives
n ̃e
n=−
k· ̃ue
ω∼
k^2 ̃φ
ω(15.64)
and so
∣∣
∣
∣̃nle ̃uhe
̃nhe ̃ule∣∣
∣
∣=
k^2 lφ ̃l
ωl
khφ ̃hk^2 h ̃φh
ωhklφ ̃l=
ωh/kh
ωl/kl>> 1 (15.65)
since the phase velocity of the high frequency mode is much higher than the phasevelocity
of the low frequency mode. Thus, the ̃nle ̃uheterm dominates and the other term can be
discarded. If the high frequency wave associated with ̃uheis an electromagnetic wave, then
the electric field is transverse so∇· ̃uhe ∼∇·E ̃h= 0in which case there is no high
frequency densityfluctuation, i.e.,n ̃he= 0.Again, only the ̃nle ̃uheterm is to be retained.
Hence, the high frequency wave equation becomes
∂^2 ̃ne
∂t^2+ ω^2 pen ̃e−3 κTe
me∇^2 n ̃e=−∂
∂t∇·
(
n ̃le ̃uhe)
. (15.66)
If the high frequency wave is electromagnetic, then∇·
(
̃nle ̃uhe)
= ̃uhe·∇ ̃nleshowing that the
non-linearity consists of a density modulation due to the high frequency motion across the
density ripples of the low frequency mode. For purposes of consistency, it isworthwhile to
express Eq.(15.66) in terms of electric field using
∇·E ̃=
1
ε 0̃neqe (15.67)so
∇·(
∂^2 E ̃
∂t^2+ ω^2 peE ̃ −
3 κTe
me∇^2 E ̃
)
=−
∂
∂t1
ε 0∇·
(
̃nleqe ̃uhe)
(15.68)
which can be integrated in space to give the general expression for a high frequency electron
plasma wave with nonlinear coupling term,
∂^2 E ̃
∂t^2+ ω^2 peE ̃ −3 κTe
me∇^2 E ̃=−
1
ε 0∂
∂t(
̃nleqe ̃uhe