Environmental Engineering Reference
In-Depth Information
5.2
Waves in Ionized Gases
5.2.1
Acoustic Oscillations
Oscillations and noise in a plasma play a much greater role than in an ordinary
gas because of the long-range nature of charged particle interactions. If a plasma
is not uniform and is subjected to external fields, a wide variety of oscillations can
occur [24, 25]. Under some conditions, these oscillations can become greatly ampli-
fied. Then the plasma oscillations affect basic plasma parameters and properties.
Below we analyze the simplest types of oscillations in a gas and in a plasma. The
natural vibrations of a gas are acoustic vibrations, that is, waves of alternating com-
pressions and rarefactions that propagate in gases. We shall analyze these waves
with the goal of finding the relationship between the frequency
ω
of the oscillation
and the wavelength
λ
, which is connected to the wave vector
k
by
j
k
jD
2
π
/
λ
.Itis
customary to refer to the amplitude
k
as the wave number.
In our analysis, we assume the oscillation amplitudes to be small. Thus, any
macroscopic parameter of the system can be expressed as
Dj
k
j
X
A
0
ω
exp[
i
(
kx
A
D
A
0
C
ω
t
)] ,
(5.47)
ω
where
A
0
is an unperturbed parameter (in the absence of oscillations),
A
0
ω
is the
amplitude of the oscillations,
is the oscillation frequency, and
k
is the appro-
priate wave number. The wave propagates along the
x
-axis. Since the oscillation
amplitude is small, an oscillation of a given frequency does not depend on oscilla-
tions with other frequencies. In other words, there is no coupling between waves
of different frequencies when the amplitudes are small. Therefore, one need retain
only the leading term in sum (5.47) and can express the macroscopic parameter
A
in the form
ω
A
0
exp[
i
(
kx
A
D
A
0
C
ω
t
)] .
(5.48)
To analyze acoustic oscillations in a gas, we can apply (5.48) to the number den-
sity of gas atoms (or molecules)
N
, the gas pressure
p
, and the mean gas velocity
w
,
and take the unperturbed gas to be at rest (
w
0
0). Using the continuity equa-
tion (4.1) and ignoring terms with squared oscillation amplitudes, we obtain
D
N
0
D
kN
0
w
0
.
ω
(5.49)
The gas velocity
w
is directed along the wave vector
for an acoustic wave (a longi-
tudinal oscillation). Similarly, the Euler equation (4.15) in the linear approximation
leads to
k
k
mN
0
p
0
,
w
0
D
ω
(5.50)
where
m
is the mass of the particles of the gas.
