In order to clarify the effects of vibrational excitation on shock-wave transitions of weak, spherical N-waves, which were generated by using sparks and exploding wires as sources, the compressible Navier-Stokes equations were solved numerically, with a vibrational-relaxation equation for oxygen. An explosion from a small pressurized sphere filled with air was used to simulate the N-waves generated from the actual sources. By employing the random-choice method (r.c.m.) with an operator-splitting technique, the effects of artificial viscosity appearing in finite-difference schemes were eliminated and accurate profiles of the shock transitions were obtained. However, a slight randomness in the variation of the shock thickness remains. It is shown that a computer simulation is possible by using a proper choice of initial parameters to obtain the variations of N-wave overpressure and half-duration with distance from the source. The calculated rise times are also shown to simulate both spark and exploding-wire data. It was found that, in addition to the vibrational-relaxation time of oxygen, both the duration (N-wave effect) and the attenuation rate (non-stationary effect) of a spherical N-wave are important factors controlling its rise time. These effects are discussed in more detail in relation to Lighthill's analytical solutions and the r.c.m. solutions for non-stationary plane waves and spherical N-waves. It is also shown that the duration and the attenuation rate of a spherical N-wave are affected by viscosity, and vibrational non-equilibrium, so that they can deviate from the results of classical, linear acoustic theory for very weak spherical waves.