Alfvén Waves in Dusty Interstellar Clouds.

N. F. Cramer, S. V. Vladimirov, PASA, 14 (2), in press.

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The Dispersion Relation

We now write the fields in terms of circularly polarized mode amplitudes:
 equation177
where the +(-) sign corresponds to the left(right) hand circularly polarized wave.

Using (11) in (7)-(10) yields, after some algebra,
 equation188

 equation196
Here
 equation206

equation218
with
equation241
In our case of tex2html_wrap_inline658 very close to 1, tex2html_wrap_inline660.

Equation (13) shows that for oblique propagation (tex2html_wrap_inline662), the amplitudes of opposite circular polarization are coupled together, i.e. the modes are not purely circularly polarized. We then obtain from (13):
 equation248

The cutoffs of tex2html_wrap_inline632 (where tex2html_wrap_inline666) correspond to
 equation253
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),
 equation258
showing, for the right hand polarized mode, the cutoff where tex2html_wrap_inline668 at tex2html_wrap_inline670, and the resonance where tex2html_wrap_inline672 at tex2html_wrap_inline674. This mode was discussed by Shukla (1992). For tex2html_wrap_inline676, we obtain the modification of the parallel propagating Alfvén 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 tex2html_wrap_inline678 at tex2html_wrap_inline680, and the left hand circularly polarized mode with tex2html_wrap_inline682 as tex2html_wrap_inline684.




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