Equations are derived which may be used to describe the propagation of electromagnetic waves in non-uniform magnetized plasma when the wave frequency is near the second electron cyclotron harmonic. The method used is to expand the linearized Vlasov equation in powers of the electron Larmor radius divided by a typical scale length. The general equations are then specialized to the problem of the coupling of transverse waves to the longitudinal modes (Bernstein modes) which exist when all quantities vary only in a plane perpendicular to a straight magnetic field. The form of these equations for two simple models of the equilibrium plasma is given. Comments are made about the equations for the higher harmonics, and the question of boundary conditions is discussed. Finally, the general equations are examined in the limit Ω→0 in order to provide equations suitable for the description of high frequency waves in non-magnetized plasmas.
