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Direct numerical simulations of turbulent pipe flow with transverse wall oscillation (WWO) and with no transverse wall oscillation (NWO) are carried out at friction Reynolds numbers Re{\tau} = 170, 360, and 720. The period and amplitude of the oscillation are selected to achieve high drag reduction in this Reynolds number range, and the effect of increasing Reynolds number on the amount of drag reduction achievable is analyzed. Of a particular interest in this study is the identification of the scales of motion most affected by drag reduction at different Reynolds numbers. To answer this question, both one-dimensional and two-dimensional spectra of different statistical quantities are analyzed with and without transverse wall oscillation. The effect of wall oscillation is found to suppress the intermediate- and large-scale motions in the buffer layer of the flow, while large-scale and very-large-scale motions in the log layer and the wake region are enhanced. While suppression of the near-wall turbulence promotes drag reduction, enhancement of the large-scale motions in the log and the wake region is found to oppose drag reduction. Since higher Reynolds number flows support development of a growing range of large-scale structures, it is suggested that their prevalence in the energy spectra combined with their negative effect on drag reduction account for a reduced effectiveness of wall oscillation as a drag reduction mechanism with increasing Reynolds numbers.