Environmental Engineering Reference
In-Depth Information
ion acoustic wave exists only in a restricted range of wave amplitudes and velocities.
Exceeding the limiting amplitude leads to a wave decaying into separate waves.
5.4.6
Parametric Instability
Nonlinear phenomena are responsible for interaction between different modes of
oscillation. A possible consequence of this interaction is the decay of a wave into
two waves. Since the wave amplitude depends on time and spatial coordinates by
the harmonic dependency exp(
i
k
r
i
ω
t
), such a decay corresponds to fulfilling
the relations
ω
D
ω
C
ω
2
,
D
k
1
C
k
2
,
(5.121)
k
0
0
1
where subscript 0 relates to the parameters of the initial wave and subscripts 1
and 2 refer to the decay waves. This instability is called parametric instability. We
consider below an example of this instability in which a plasma oscillation decays
into a plasma oscillation of a lower frequency and an ion acoustic wave (ion sound).
The electric field of the initial plasma oscillation is
E
D
E
0
cos(
k
0
x
ω
0
t
),
where
x
is the direction of propagation. In the zero approximation we assume
the electric field amplitude
E
0
and other wave parameters to be real values. The
equation of motion for electrons
m
e
d
v
0
/
dt
D
eE
yields the electron velocity
0
).
Let another plasma wave and the ion sound wave be excited in the system si-
multaneously with the initial plasma oscillation, and let their amplitudes be small
compared with the amplitude of the initial oscillation. Consider the temporal de-
velopment of these waves taking into account their interaction with each other and
with the initial oscillation. Since ion velocities are much lower than electron ve-
locities, one can analyze these waves separately. The equation of motion and the
continuity equation for ions are
D
u
0
cos(
k
0
x
ω
0
t
), where
u
0
D
eE
0
/(
m
e
ω
v
0
N
i
@
m
i
d
@
N
0
@
v
i
@
v
i
dt
D
eE
,
C
D
0.
t
x
Here
m
i
is the ion mass,
v
i
is the ion velocity,
N
0
is the equilibriumnumber density
of ions,
N
i
is the perturbation of the ion number density due to the oscillation,
and
E
is the electric field due to the oscillations. Elimination of the ion velocity
from these equations yields
2
N
i
@
@
eN
0
m
i
@
E
@
C
D
0 .
(5.122)
t
2
x
One can find the electric field strength from the equation of motion for the elec-
trons by averaging over fast oscillations. The one-dimensional Euler equation (4.6)
