Dispersion relations for ion cyclotron oscillations

The natural modes of oscillation of a cold plasma, in the presence of a longitudinal external magnetic field, are examined near the ion cyclotron frequency. It is shown that under appropriate conditions, both the ordinary and the extraordinary hydrodynamic waves can be modified, giving modes whose natural frequency is in the neighbourhood of the ion cyclotron frequency. Dispersion relations for these modes are given in the case of cylindrical geometry, and the excitation of oscillations close to the ion gyrofrequency by a small-density beam is discussed.