We examine the impact of baryon-dark matter relative velocities on intergalactic small-scale structure and the 21 cm signal during reionization. Streaming velocities reduced clumping in the intergalactic medium (IGM) on mass scales of $sim 10^4 - 10^8$ M$_{odot}$. This effect produced a distinct baryon acoustic oscillation (BAO) feature in the 21 cm power spectrum at wave numbers $ksim 0.1$ h/Mpc, near which forthcoming surveys will be most sensitive. In contrast to the highly uncertain impact of streaming velocities on star formation, the effect on clumping is better constrained because it is set mainly by cosmology and straightforward gas dynamics. We quantify the latter using coupled radiation-hydrodynamic simulations that capture the Jeans scale of pre-reionization gas. The clumping factor of ionized gas is reduced by 5-10% in regions with RMS streaming velocities. The suppression peaks $approx 5$ Myr after a region is reionized, but disappears within 200 Myr due to pressure smoothing. We model the corresponding impact on the 21 cm signal and find that the BAO feature is most likely to appear at $approx$ 10 % ionization. During this phase, the feature may appear at the 1 % (5 %) level at $k sim 0.1 (0.06)$ h/Mpc with an amplitude that varies by a factor of $< 10$ across a range of reionization histories. We also provide a model for the signal originating from streaming velocitys impact on ionizing sources, which can vary by 4 orders of magnitude depending on highly uncertain source properties. We find that the clumping signal probably dominates the source one unless Population III star formation in $10^6 - 10^8$ M$_{odot}$ halos contributed significantly to the first 10% of reionization.
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