Fundamentals of Plasma Physics

114 Chapter 3. Motion of a single plasma particle

3.9.4 Wave-particle energy transfer in a sinusoidal wave

The calculation will now redone for the physically more relevant situationwhere a group
of particles interact with a sinusoidal wave. As a prerequisite for doingthis calculation it
must first be recognized that two distinct classes of particles exist, namely those which are
trapped in the wave and those which are not. The trajectories of trapped particles differs
in a substantive way from untrapped particles, but for low amplitude waves the number of
trapped particles is so small as to be of no consequence. It therefore will be assumed that
the wave amplitude is sufficiently small that the trapped particles can be ignored.
Particle energy is conserved in the wave frame but not in the lab frame because the
particle Hamiltonian is time-independent in the wave frame but not inthe lab frame. Since
each additional conserved quantity reduces the number of equations to be solved, itis
advantageous to calculate the particle dynamics in the wave frame, and then transform
back to the lab frame.
The analysis in Sec.3.9.2 of particle motion in a sawtooth potential showed that ran-
domly phased groups of particles have their average velocity slow down, i.e., the average
velocity of the group tends towards zero as observed in the frame of the sawtooth potential.
If the sawtooth potential were moving with respect to the lab frame, the sawtooth poten-
tial would appear as a propagating wave in the lab frame. A lab-frame observer would see
the particle velocities tending to come to rest in the sawtooth frame,i.e., the lab-frame av-
erage of the particle velocities would tend to converge towards the velocity with which the
sawtooth frame moves in the lab frame.
The quantitative motion of a particle in a one-dimensional wave potentialφ(x,t) =
φ 0 cos(kx−ωt)will now be analyzed in some detail. This situation corresponds to a
particle being acted on by a wave traveling in the positivexdirection with phase velocity
ω/k.It is assumed that there is no magnetic field so the equation of motionis simply

dv

dt

=

qkφ 0

m

sin(kx−ωt). (3.219)

Att= 0the particle’s position isx=x 0 and its velocity isv=v 0 .The wave phase at the
particle location is defined to beψ=kx−ωt. This is a more convenient variable thanx
and so the differential equations forxwill be transformed into a corresponding differential
equation forψ. Usingψas the dependent variable corresponds to transforming to the wave
frame, i.e., the frame moving with the phase velocityω/k,and makes it possible to take
advantage of the wave-frame energy being a constant of the motion. The equations are less
cluttered with minus signs if a slightly modified phase variableθ=kx−ωt−πis used.
The first and second derivatives ofθare
dθ
dt

=kv−ω (3.220)

and
d^2 θ
dt^2
=k

dv

dt

. (3.221)

Substitution of Eq.(3.221) into Eq.(3.219) gives

d^2 θ

dt^2

+

k^2 qφ 0

m

sinθ= 0. (3.222)