  Buneman instability is often driven in magnetic reconnection. Understanding
how velocity shear in the beams driving the Buneman instability affects the
growth and saturation of waves is relevant to turbulence, heating, and
diffusion in magnetic reconnection. Using a Mathieu-equation analysis for weak
cosine velocity shear together with Vlasov simulations, the effects of shear on
the kinetic Buneman instability are studied in a plasma consisting of strongly
magnetized electrons and cold unmagnetized ions. In the linearly unstable
phase, shear enhances the coupling between oblique waves and the sheared
electron beam, resulting in a wider range of unstable eigenmodes with common
lower growth rates. The wave couplings generate new features of the electric
fields in space, which can persist into the nonlinear phase when electron holes
form. Lower hybrid instabilities simultaneously occur at
$k_{\shortparallel}/k_{\perp} \sim \sqrt{m_e/m_i}$ with a much lower growth
rate, and are not affected by the velocity shear.
