The authors consider the penetration of electromagnetic waves with a frequency of the order of the ion cyclotron frequencies and with a fixed longitudinal wave number k||, so that N|| = k||cω ≫ 1 deep into an inhomogeneous plasma with two ion species. The propagation of two kinds of waves (fast and slow) with widely differing polarization and transverse refraction index is possible. For both types of waves there is an evanesence region at the plasma periphery.The evanescence region is narrow for slow waves and the y easily penetrate the plasma. In a dense plasma they become electrostatic and can reach the ion-ion hybrid resonance region. However, the damping of these waves due to Cherenkov interaction with electrons in a high-temperature plasma is strong and therefore they are not suitable for heating plasma of large dimensions, as they are absorbed at the plasma periphery.The fast waves have a wider evanescence region and can be excited effectively only if N2 is not too high. These waves can be completely absorbed in the plasma (due to Cherenkov interaction with electrons) if where a is the plasma radius and . Fast waves can also reach the region where they are transformed into slow waves. In this region their damping increases considerably. It is shown that the transformation region in an inhomo-geneous plasma with two ion species in a non-uniform magnetic field may be at the centre of the plasma. Fast waves can be used effectively for heating plasma of large dimensions.