Ion sound wave
In a plasma, electromagnetic interactions allow
compressional longitudinal waves to propagate
at lower temperatures and pressures
than would be possible in a neutral gas.
We start from the two-fluid model’s momentum equations,
rewriting the electric field
and the pressure gradient ,
and arguing that because :
Note that we neglect ion-electron collisions,
and allow for separate values of .
We split , , and into an equilibrium
(subscript ) and a perturbation (subscript ):
Where the perturbations , , and are tiny,
and the equilibrium components , , and
are assumed to satisfy:
Inserting this decomposition into the momentum equations yields new equations:
Using the assumed properties of , , and ,
and discarding products of perturbations for being too small,
we arrive at the below equations.
Our choice lets us linearize
the material derivative
for the ions:
Because we are interested in linear waves,
we make the following plane-wave ansatz:
Which we then insert into the momentum equations for the ions and electrons:
The electron equation can easily be rearranged
to get a relation between and :
Due to their low mass, the electrons’ heat conductivity
can be regarded as infinite compared to the ions’.
In that case, all electron gas compression is isothermal,
meaning it obeys the ideal gas law , so that .
Note that this yields the first-order term of a Taylor expansion
of the Boltzmann relation.
At equilibrium, quasi-neutrality demands that ,
so we can rearrange the above relation to ,
which we insert into the ion equation to get:
Where we have taken the dot product with ,
and used that .
In order to simplify this equation,
we turn to the two-fluid ion continuity relation:
Into which we insert our plane-wave ansatz,
and substitute as before, yielding:
Substituting this in the ion momentum equation
leads us to a dispersion relation :
Finally, we would like to find an expression for .
It cannot be , because then could not be nonzero,
according to Gauss’ law.
Nevertheless, some authors tend to ignore this fact,
thereby making the so-called plasma approximation.
We will not, and thus turn to Gauss’ law:
One final time, we insert our plane-wave ansatz,
and use our Boltzmann-like relation between and
to substitute :
Where is the electron Debye length.
We thus reach the following dispersion relation,
which governs ion sound waves or ion acoustic waves:
The aforementioned plasma approximation is valid if ,
which is often reasonable,
in which case this dispersion relation reduces to:
The phase velocity of these waves,
i.e. the speed of sound, is then given by:
Curiously, unlike in a neutral gas,
this velocity is nonzero even if ,
meaning that the waves still exist then.
In fact, usually the electron temperature dominates ,
even though the main feature of these waves
is that they involve ion density fluctuations .
- F.F. Chen,
Introduction to plasma physics and controlled fusion,
3rd edition, Springer.
- M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,