Fundamentals of Plasma Physics

3.9 Wave-particle energy transfer 115

By defining thebouncefrequency

ω^2 b=

k^2 qφ 0

m

(3.223)

and the dimensionless bounce-normalized time

τ=ωbt, (3.224)

Eq. (3.222) reduces to the pendulum-like equation

d^2 θ

dτ^2

+ sinθ= 0. (3.225)

Upon multiplying by the integrating factor2dθ/dτ,Eq.(3.225) becomes

d

dτ

[(

dθ

dτ

) 2

−2cosθ

]

= 0. (3.226)

This integrates to give
(
dθ
dτ

) 2

−2cosθ=λ=const. (3.227)

which indicates the expected energy conservation in the wave frame. Thevalue ofλis
determined by two initial conditions, namely the wave-frame injection velocity
(
dθ
dτ

)

τ=0

=

1

ωb

(

dθ

dt

)

t=0

=

kv 0 −ω

ωb

≡α (3.228)

and the wave-frame injection phase

θτ=0=kx 0 −π≡θ 0. (3.229)

Inserting these initial values in the left hand side of Eq. (3.227) gives

λ=α^2 −2cosθ 0. (3.230)

Except for a constant factor,

• λis the total wave-frame energy


• (dθ/dτ)^2 is the wave-frame kinetic energy

•−2cosθis the wave-frame potential energy.
If− 2 < λ < 2 ,then the particle is trapped in a specific wave trough and oscillates
back and forth in this trough. However, ifλ > 2 ,the particle is untrapped and travels
continuously in the same direction, speeding up when traversing a potential valley and
slowing down when traversing a potential hill.
Attention will now be restricted to untrapped particles with kineticenergy greatly ex-
ceeding potential energy. For these particles

α^2 >> 2 (3.231)