With the full time-domain electromagnetic simulation technique, the time-domain propagation properties such as the temporal pulse waveform, the pulse compression and the electric field distribution inside the material are comparably investigated by transmitting the time-reversal pulsed electromagnetic wave and the pulsed electromagnetic wave through the double negative material of the Smith structure, respectively. The results show that after the time-reversal pulsed electromagnetic wave passes through the double negative material, the received electromagnetic wave exhibits remarkable spatial and temporal focusing at the location of the initial excitation. The most interesting physical phenomenon is that the peak values of the electric field inside the double negative material become smaller and tend to be uniform in the magnitude. These novel physical phenomena would be very helpful for developing new electronic devices and systems with double negative materials for the purpose of high power applications.