A study has been carried out on the build-up and characterization of nuclear burn-up wave in fast neutron multiplying media. The focus of the study is more on the transient part of the burn-up wave though the steady state is also studied. The characteristics of the transient part are expressed in terms of new parameters, i.e., Transient Length (TL) and Transient Time (TT) elapsed in establishing the asymptotic burn-up wave. The TT and TL are defined, respectively, as the time and length needed to establish 95% and 99% of the asymptotic neutron flux propagating in the media. From these parameters, the transient wave velocity is determined. The characteristics of asymptotic part of the wave are determined in terms of wave velocity, Full Width at Half Maximum (FWHM) and Full Width at 10% Maximum (FW10M) of neutron flux distribution in the space. A parametric study is also carried out to investigate the sensitivity of these parameters to some of the physical parameters of the ignition zone and the breeder zone of the reactor. The characterization parameters would be very meaningful to understand the transient characteristics of the self-sustaining nuclear burn-up wave and in evaluating the quality of the wave by the researchers working in the field of nuclear burn-up wave build-up and propagation.