A new model is developed to simulate the space-time evolution of a propagating coherent whistler mode wave pulse in the magnetosphere. The model is applied to the case of single frequency (2--5 kHz) wave pulses injected into the magnetosphere near L≂4, using the VLF transmitting facility at Siple Station, Antarctica. The mechanism for growth is cyclotron resonance between the circularly polarized waves and the gyrating energetic electrons of the radiation belts. Application of this model reproduces observed exponential wave growth up to a saturated level. Additional, the model predicts the observed initial linear increase in the output frequency versus time. This is the first time these features have been reproduced using applied wave intensities small enough to be consistent with satellite measurements. The center velocities of the electrons entering the wave pulse are selected in a way which maximizes the growth rate. The results show the importance of the transient aspects in the wave growth process. The growth established as the wave propagates toward the geomagnetic equator results in a spatially advancing wave phase structure due mainly to the geomagnetic inhomogeneity. Through the feedback of this radiation upon other electrons, conditions are established which result in a linearly increasing output frequency with time. ¿ American Geophysical Union 1990