We present numerical simulation results for the propagation of Alfvén waves in the charge starvation regime. This is the regime where the plasma density is below the critical value required to supply the current for the wave. We analyse a conservative scenario where Alfvén waves pick up charges from the region where the charge density exceeds the critical value and advect them along at a high Lorentz factor. The system consisting of the Alfvén wave and charges being carried with it, which we call charge-carrying Alfvén wave (CC-AW), moves through a medium with small, but non-zero, plasma density. We find that the interaction between CC-AW and the stationary medium has a two-stream like instability which leads to the emergence of a strong electric field along the direction of the unperturbed magnetic field. The growth rate of this instability is of the order of the plasma frequency of the medium encountered by the CC-AW. Our numerical code follows the system for hundreds of wave periods. The numerical calculations suggest that the final strength of the electric field is of the order of a few per cent of the AW amplitude. Little radiation is produced by the sinusoidally oscillating currents associated with the instability during the linear growth phase. However, in the non-linear phase, the fluctuating current density produces strong EM radiation near the plasma frequency and limits the growth of the instability.