  A thory of field-induced domain wall (DW) propagation is developed. The
theory not only explains why a DW in a defect-free nanowire must propagate at a
finite velocity, but also provides a proper definition of DW propagation
velocity. This definition, valid for an arbitrary DW structure, allows one to
compute the instantaneous DW velocity in a meaningful way even when the DW is
not moving as a rigid body. A new velocity-field formula beyond the Walker
breakdown field, which is in excellent agreement with both experiments and
numerical simulations, is derived.
