Results of high-resolution optical spectroscopy of V404 Cyg (the optical counterpart of the X-ray transient GS2023 + 338), conducted 13 months after the May 1989 outburst are presented. The double-peaked H-alpha line exhibits a strong S-wave modulation from which a periodicity of 5.7 hr is deduced. The level of the continuum is also modulated on the same time-scale. This behavior is identified with the orbital period of the system. There are also changes in the FWHM and peak separation, which evolves from widening to subsequent shrinking. Such variations suggest a very complex line profile consisting perphaps of several different components.