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)