The process of earthquake generation is governed by a coupled non-linear system consisting of the equation of motion in elastodynamics and a fault constitutive relation. On the basis of the results of stick-slip experiments we constructed a theoretical source model with a slip-dependent constitutive law. Using the theoretical source model, we simulated the transition process numerically from quasi-static nucleation to high-speed rupture propagation and succeeded in quantitatively explaining the three phases observed in stick-slip experiments, that is very slow (1 cm s−1 ) quasi-static nucleation preceding the onset of dynamic rupture, dynamic but slow (10 m s−1 ) rupture growth without seismic-wave radiation, and subsequent high-speed (2 km s−1 ) rupture propagation. Theoretical computation of far-field waveforms with this model shows that a slow initial phase preceding the main P phase expected from a classical source model is radiated in the accelerating stage from the slow dynamic rupture growth to the high-speed rupture propagation. On the assumption that the physical law governing rupture processes in natural earthquakes is essentially the same as that in stick-slip events, we scaled the theoretical source model explaining the stick-slip experiments to the case of natural earthquakes so that the scaled source model explains the observed average stress drop, the critical nucleation-zone size, and the duration of the slow initial phase well. The physical parameters prescribing the source model are the weak-zone size L , the critical weakening displacement D¯c , the breakdown strength drop τ¯b , and the rigidity μ of the surrounding elastic medium. In scaling these parameters, we held a non-dimensional controlling parameter μ′ = (μD¯c )/(τ¯b L ) in numerical simulation constant. From the results of scaling we found the following fundamental relations between the source parameters: (1) the critical weakening displacement D¯c is in proportion to the weak-zone size L , but (2) the breakdown strength drop τ¯b is independent of L .