The paper considers receiving acoustic horns designed for particle velocity amplification and suitable for use in vector sensing applications. Unlike conventional horns, designed for acoustic pressure amplification, acoustic velocity horns (AVHs) deliver significant velocity amplification even when the overall size of the horn is much less than an acoustic wavelength. An AVH requires an open-ended configuration, as compared to pressure horns which are terminated at the throat. The appropriate formulation, based on Webster's one-dimensional horn equation, is derived and analyzed for single conical and exponential horns as well as for double-horn configurations. Predicted horn amplification factors (ratio of mouth-to-throat radii) were verified using numerical modeling. It is shown that three independent geometrical parameters principally control a horn's performance: length l, throat radius R(1), and flare rate. Below a predicted resonance region, velocity amplification is practically independent of frequency. Acoustic velocity horns are naturally directional, providing maximum velocity amplification along the boresight.