Abstract: A high electric field gradient electron accelerator utilizing short duration, microwave radiation, and capable of operating at high field gradients for high energy physics applications or at reduced electric field gradients for high average current intermediate energy accelerator applications. Particles are accelerated in a smooth bore, periodic undulating waveguide, wherein the period is so selected that the particles slip an integral number of cycles of the r.f. wave every period of the structure. This phase step of the particles produces substantially continuous acceleration in a traveling wave without transverse magnetic or other guide means for the particle.