The expected event rate of ¯ ν$_{e}$ in the case of the inverted mass hierarchy and that of ν$_{e}$ in the normal mass hierarchy depend on the mixing angle θ$_{13}$ . When sin $^{2}$2 θ$_{13}$ is larger than ∼ 10 − $^{4}$, the influence of the shock wave appears in the observation after 2 s in our model. It is the time when the shock wave reaches the H-resonance. Therefore, the shock effect and the constraint on θ$_{13}$ can be inferred even by the event rate integrated by the whole energy range. However, it is difficult to distinguish the influence of the shock wave and the neutrino oscillation parameters only from the time evolution of the event rate. The time-integrated ratio of the events of high-energy to low-energy neutrinos is another indicator to examine the shock effect and constrain mass hierarchy and θ$_{13}$ . The parame- ter dependence of the time integrated ratio R$_{x}$ (x = SK or SNO) could be an important observational signature to discriminate neutrino mass hierarchy, normal or inverted, as well as the mixing angle, θ$_{13}$ . Both of the ratios R$_{SK}$ and R$_{SNO}$ correlate with the value of θ$_{13}$ in inverted mass hierarchy. Therefore, the neutrino oscillation parameters θ$_{13}$ and the mass hierarchy, might be able to be constrained by this ratio. The shock effect reduces the ratios R$_{SK}$ and R$_{SNO}$ . This effect is not distinguished with smallness of θ$_{13}$ . The time-dependent ratio of high- to low-energy neutrino events is more preferable indi- cator to find out the shock effect clearly. The ratio decreases after 2 s for R$_{SNO}$ ( t ) in normal hierarchy and for R$_{SK}$ ( t ) in inverted hierarchy. The decrease is the most remarkable in the case of sin $^{2}$2 θ$_{13}$ = 10 − 3 and more clearly in inverted hierarchy. The dramatic decrease in the ratio could be a clear signal for the shock wave effect and would constrain the minimum value of sin $^{2}$2 θ$_{13}$ . Therefore, observations of the time-dependent ratio of the high- to low- energy neutrino event are an important observable to find the effects of shock propagation in supernovae. We would like to thank K. Kotake for fruitful discussions. SK thanks to T. Takiwaki and S. Harikae for help comments on the numerical scheme. This research work was supported Acknowledgments 22