Characteristic waves in a plasma What does "characteristic wave" mean in the context of plasma waves? For example, when propagating parallel to the plasma magnetic field, the characteristic wave is circularly polarized. Does this mean that only a circularly polarized wave can propagate in this direction? Could an elliptically polarized wave travel in this direction instead?
 A: In plasma physics there is no particular usage of the term "characteristic waves". However, from looking at Hutchinson book it becomes clear that he derives this name from characteristic polynomial. In mathematics, the characteristic polynomial is the polynomial one needs to solve to obtain eigenvalues of a system, then compute eigen vector of the system.
To study wave modes in plasma, one needs to find the eigenvalues of dielectric tensor (which are the dispersion relations of the different wave modes) and the eigen vectors (which are the electric fields associated with those wave modes). Since dispersion relations of wave modes are obtained by solving the characteristic polynomial of dielectric tensor, the name "characteristic waves" is not very strange.
In cold magnetized plasma there are 5 possible solutions of characteristic polynomial:


*

*One solution represents the case when electric field (E) and wave vector (k) are both parallel to background magnetic field (B0), this is called plasma oscillation (k can't be defined for cold plasma in this case).

*Two solutions when E is normal to B0, and k is parallel to B0, those are called L wave and R wave.

*Two solutions when E and k are both normal to B0, those are called X wave and O wave.
The case you are referring to in your example describes the R wave and L wave. They are the characteristic waves because they are the solutions of the characteristic polynomial that describe the case in which k is parallel to B. I hope I made it clear.
With respect to circular versus elliptical. The R and L waves are circularly polarized because the dielectric tensor is symmetric, which is a direct consequence of the 
symmetry in magnetic field terms in equations of motion. Please refer to equations 4.1.17 in Hutchinson and follow the derivation until you arrive to equation 4.1.20 where you can clearly see the symmetry in conductivity tensor. 
To break the symmetry, the magnetic field terms should be different. I personally don't know a reason of a case where they are different. I did a quick search for such a case in literature and I couldn't find any result on such a case, which made me believe if it existed it would be in exceptional circumstances. Otherwise one would have found it easily from the first few pages of the search.
I hope that helped!!
A: First of all, there is no such a term "characteristic wave" in physics, certainly it is not in use in plasma physics. There is a large variety of waves that can propagate in plasma. Along the magnetic field, it can be a magnetosonic wave or a Langmuir wave which are of longitudinal polarization; or it can be transverse modes like the shear-Alfven wave (SAW). Combining two SAW linearly polarized in perpendicular directions, one can produce a circularly polarized SAW, if the  amplitudes of the two SAW are equal and the phase shift is 90 degrees. In general, for different amplitudes or a different phase shift, an elliptically polarized SAW would result.
However, that equation in Hutchinson discusses specifically high frequency waves (RF) that couple to the electron gyro-motion. For these waves the phase speed is different depending on whether the mode in right- or left-polarized. Therefore, opposite to the case of a SAW or an EM wave in vacuum, a linear combination of R and L waves is not an eigenmode, so elliptically and linearly polarized waves are not possible in this context. If you initiate an elliptically polarized perturbation in such a system it will split into R and L waves traveling separately. 
