During the Voyager 2 flyby of Uranus, strong electron plasma oscillations (Langmuir waves) were detected by the plasma wave instrument in the 1.78-kHz channel on January 23--24, 1986, prior to the inbound bow shock crossing. Langmuir waves are excited by energetic electrons streaming away from the bow shock. The goal of this work is to estimate the location and motion of Uranus' bow shock using Langmuir wave data, together with the spacecraft positions and the measured interplanetary magnetic field. The following three remote sensing analyses were performed: the basic remote sensing method, the lag time method, and the trace-back method. Because the interplanetary magnetic field was highly variable, the first analysis encountered difficulties in obtaining a realistic estimation of Uranus' bow shock motion. In the lag time method developed here, time lags due to the solar wind's finite convection speed are taken into account when calculating the shock's standoff distance. In the new trace-back method, limits on the standoff distance are obtained as a function of time by reconstructing electron paths. Most of the results produced by the latter two analyses are consistent with predictions based on the standard theoretical model and the measured solar wind plasma parameters. Differences between our calculations and the theoretical model are discussed. ¿ American Geophysical Union 1996