This oscillation is closely related to the plasma oscillation found in unmagnetized plasmas or parallel to the magnetic field, where the ωpe term arises from the electrostatic Coulomb restoring force and the 3k²ve,th² term arises from the restoring force of electron pressure.
Consider a plane wave where all perturbed quantities vary as exp(i(kx-ωt)).
If the displacement in the direction of propagation is δx, then Thus the perpendicular magnetic field effectively provides a harmonic restoring force with a frequency ωce, explaining the third term in the dispersion relation.
The particle orbits (or fluid trajectories) are ellipses in the plane perpendicular to the magnetic field, elongated in the direction of propagation.
For propagation at angles oblique to the magnetic field, two modes exist simultaneously.