Non-linear periodic waves propagating across a magnetic field are investigated, taking relativistic effects into account. The governing equations of the phenomena are taken to be the relativistic two-fluid plasma equations for a cold plasma together with Maxwell's equations. The analytical solution in the non-relativistic case was obtained by Adlam and Allen and others, a case where the quasi- neutral approximation is a good one and the inertia of the electrons plays an essential role in the dispersion effect of the plasma.

In contrast, the analysis of the relativistic waves shows that the quasi- neutral approximation becomes invalid and that the charge separation plays an essential role in the dispersion effect while the inertia of the electrons plays minor role only. Therefore the phenomena to be investigated have a length scale of the appropriately defined Debye distance (the ion Larmor radius), while in the non-relativistic case the width of the wave is of the order of the geometric mean of the Larmor radii of the electrons and of the ions. In the present case where the electron mass may be neglected, there exists an exact solution which is a good approximation for a real plasma. The solitary wave solution is found as the limiting case of infinite periodicity.