Dielectric haloscopes: sensitivity to the axion dark matter velocity

We study the effect of the axion dark matter velocity in the recently proposed dielectric haloscopes, a promising avenue to search for well-motivated high mass ($40-400~\mu$eV) axions. We describe non-zero velocity effects for axion-photon mixing in a magnetic field and for the phenomenon of photon emission from interfaces between different dielectric media. As velocity effects are only important when the haloscope is larger than about 20% of the axion de Broglie wavelength, for the planned MADMAX experiment with 80 dielectric disks the velocity dependence can safely be neglected. However, an augmented MADMAX or a second generation experiment would be directionally sensitive to the axion velocity, and thus a sensitive measure of axion astrophysics.