  We study the directional effect of the expected axion dark matter signal in a
resonant cavity of an axion haloscope detector, for cavity geometries not
satisfying the condition that the axion de Broglie wavelength $\lambda_a$ is
sufficiently larger than the cavity dimensions $L$ for a fully coherent
conversion, i.e. $\lambda_a \gtrsim 2\pi L$. We focus on long thin cavities
immersed in dipole magnets and find, for appropriately chosen cavity lengths,
an O(1) modulation of the signal with the cavity orientation with respect the
momentum distribution of the relic axion background predicted by the isothermal
sphere model for the galactic dark matter halo. This effect can be exploited to
design directional axion dark matter detectors, providing an unmistakable
signature of the extraterrestrial origin of a possible positive detection.
Moreover, the precise shape of the modulation may give information of the
galactic halo distribution and, for specific halo models, give extra
sensitivity for higher axion masses.
