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We present an analysis of the kinematics of the Radcliffe Wave, a 2.7-kpc-long sinusoidal band of molecular clouds in the solar neighborhood recently detected via 3D dust mapping. With Gaia DR2 astrometry and spectroscopy, we analyze the 3D space velocities of $\sim 1500$ young stars along the Radcliffe Wave in action-angle space, using the motion of the wave's newly born stars as a proxy for its gas motion. We find that the vertical angle of young stars -- corresponding to their orbital phase perpendicular to the Galactic plane -- varies significantly as a function of position along the structure, in a pattern potentially consistent with a wave-like oscillation. This kind of oscillation is not seen in a control sample of older stars from Gaia occupying the same volume, disfavouring formation channels caused by long-lived physical processes. We use a ``wavy midplane'' model to try to account for the trend in vertical angles seen in young stars, and find that while the best-fit parameters for the wave's spatial period and amplitude are qualitatively consistent with the existing morphology defined by 3D dust, there is no evidence for additional velocity structure. These results support more recent and/or transitory processes in the formation of the Radcliffe Wave, which would primarily affect the motion of the wave's gaseous material. Comparisons of our results with new and upcoming simulations, in conjunction with new stellar radial velocity measurements in Gaia DR3, should allow us to further discriminate between various competing hypotheses.