Weakly relativistic dispersion of Bernstein waves

Weakly relativistic effects on the dispersion of Bernstein waves are investigated for waves propagating nearly perpendicular to a uniform magnetic field in a Maxwellian plasma. Attention is focused on those large-wave-vector branches that are either weakly damped or join continuously onto weakly damped branches since these are the modes of most interest in applications. The transition between dispersion at perpendicular and oblique propagation is examined and major weakly relativistic effects on dispersion at perpendicular propagation are found to persist at other angles; these effects can dominate even in low-temperature plasmas. A number of simple analytic criteria are obtained which delimit the ranges of harmonic number and propagation angle within which various types of weakly damped Bernstein modes can exist.