The Dispersion Relation

We now write the fields in terms of circularly polarized mode amplitudes:
á 
where the +(-) sign corresponds to the left(right) hand circularly polarized wave.

Using (11) in (7)-(10) yields, after some algebra,
á 

á 
Here
á 


with

In our case of very close to 1, .

Equation (13) shows that for oblique propagation (), the amplitudes of opposite circular polarization are coupled together, i.e. the modes are not purely circularly polarized. We then obtain from (13):
á 

The cutoffs of (where ) correspond to
á 
which is the parallel propagation case treated by Pilipp et al. (1987). For no collisions, a case previously treated by Shukla (1992) and Mendis and Rosenberg (1992), the parallel propagation dispersion relation is, from (18),
á 
showing, for the right hand polarized mode, the cutoff where at , and the resonance where at . This mode was discussed by Shukla (1992). For , we obtain the modification of the parallel propagating Alfvö©n wave due to dust discussed by Vladimirov and Cramer (1996) as the basis for a discussion of nonlinear effects: the right hand circularly polarized mode with a cutoff in at , and the left hand circularly polarized mode with as .

á© Copyright Astronomical Society of Australia 1997