Environmental Engineering Reference
In-Depth Information
Figure 5.8
Dispersion relation (5.54) for air at
T
D
300 K in the centimeter range of wave-
lengths.
Because
D
/(
c
p
N
)
v
T
/
λ
and
c
s
v
T
,where
λ
is the free path length of gas
molecules and
v
T
is a typical thermal velocity of the molecules, condition (5.55)
mayberewrittenintheform
λ
k
1.
That is, the wavelength of the oscillations is long compared with the mean free
path of the molecules of the gas.
5.2.2
Plasma Oscillations
Plasma oscillations which result from long-range interaction of electrons as a light
charged plasma component are weakly excited oscillations in ionized gases [26-28].
Plasma oscillations are the simplest oscillations for a plasma, and the frequency of
such oscillations of an infinite wavelength is given by (1.13). Below we derive the
dispersion relation for these oscillations of finite wavelengths. Then we assume
the ions to be at rest and their charges to be uniformly distributed over the plas-
ma volume. Like to the case of acoustic oscillations in a gas, we shall derive the
dispersion relation for the plasma waves from the continuity equation (4.1), the
Euler equation (4.6), and the adiabatic condition (5.55) for the wave. Moreover, we
take into account that the motion of electrons produces an electric field owing to
the disturbance of the plasma quasineutrality. The electric field term is introduced
into the Euler equation (4.6), and the electric field strength is determined by the
Poissonequation(1.4).
As with the deduction of the dispersion relation for acoustic oscillations, we
assume that the parameters of the oscillating plasma can be written in the
form (5.48), and in the absence of oscillations both the mean electron velocity
w
