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Ion Sound Waves
If we now take ion dynamics into account then the dispersion relation (1007),
for electrostatic plasma waves, generalizes to

(1032) 
i.e., we simply add an extra term for the ions which has an
analogous form to the electron term. Let us search for a wave with
a phase velocity, , which is much less than the electron thermal
velocity, but much greater than the ion thermal velocity. We may assume
that
for the ion term, as we did previously for the
electron term. It follows that, to
lowest order, this term reduces to
. Conversely, we
may assume that
for the electron term. Thus, to
lowest order we may neglect in the velocity space
integral. Assuming
to be a Maxwellian with temperature , the electron term reduces to

(1033) 
Thus, to a first approximation, the dispersion relation can be written

(1034) 
giving

(1035) 
For
, we have
, a dispersion relation
which is like that of an ordinary sound wave, with the pressure provided by the
electrons, and the inertia by the ions. As the wavelength is reduced towards the
Debye length, the frequency levels off and approaches the ion plasma
frequency.
Let us check our original assumptions. In the long wavelength limit, we see that
the wave phase velocity
is indeed much less than the
electron thermal velocity [by a factor
], but that it
is only much greater than the ion thermal velocity if the ion temperature, ,
is much less than the electron temperature, . In fact, if
then the wave phase velocity can lie on almost flat portions of the
ion and electron distribution functions, as shown in Fig. 37, implying that
the wave is subject to
very little Landau damping. Indeed, an ion sound wave can only propagate a distance of order its wavelength
without being strongly damped provided that is at least five to ten times greater than .
Of course, it is possible to obtain the ion sound wave dispersion relation,
, using fluid theory. The kinetic treatment used here
is an improvement on the fluid theory to the extent that no equation of
state is assumed, and it makes it clear to us that ion sound waves are subject to
strong Landau damping (i.e., they cannot be considered normal modes of the
plasma) unless .
Figure 37:
Ion and electron distribution functions with .

Next: Waves in Magnetized Plasmas
Up: Waves in Warm Plasmas
Previous: Plasma Dispersion Function
Richard Fitzpatrick
20110331