To study the large-amplitude Alfv¿n waves that are frequently observed in the solar wind, the nonlinear equations of electron and ion fluids coupled to Maxwell equations have been solved to obtain the relationships between the fluid density and magnetic field for finite amplitude Alfv¿n waves in the cold plasma limit. Two regimes of the parameter &ohgr;c/&OHgr; (ratio of characteristic frequency to the ion Larmor frequency) are examined. When &ohgr;c/&OHgr;=0, we obtain an exact simple wave solution for the linearly polarized Alfv¿n wave, and recover the earlier results of [Montgomery, 1959a>. For finite &ohgr;c/&OHgr;, the geometric optics approximation which is valid for large wave numbers yields a new relationship which involves elliptically polarized Alfv¿n waves. Both of these waves can be shown to evolve into shocks. ¿ American Geophysical Union 1995