  Synthetic spin-orbit (SO) coupling, an important ingredient for quantum
simulation of many exotic condensed matter physics, has recently attracted
considerable attention. The static and dynamic properties of a SO coupled
Bose-Einstein condensate (BEC) have been extensively studied in both theory and
experiment. Here we numerically investigate the generation and propagation of a
\textit{dynamical} spin-density wave (SDW) in a SO coupled BEC using a fast
moving Gaussian-shaped barrier. We find that the SDW wavelength is sensitive to
the barrier's velocity while varies slightly with the barrier's peak potential
or width. We qualitatively explain the generation of SDW by considering a
rectangular barrier in a one dimensional system. Our results may motivate
future experimental and theoretical investigations of rich dynamics in the SO
coupled BEC induced by a moving barrier.
