dominated reactions for SK and SNO detectors are the ¯ ν$_{e}$ -induced charged current reactions; ¯ ν$_{e}$ + p → e $^{+}$+ n for SK and ¯ ν$_{e}$ + d → n + n + e + for SNO. Both ratios R$_{SK}$ and R$_{SNO}$ increases with the mixing angle θ$_{13}$ . For a given θ$_{13}$ , the ratios in the case with shock are smaller than those without the shock. On the other hand, in the normal hierarchy, the ratio shows a small dependence on sin $^{2}$2 θ$_{13}$ for R$_{SK}$ and R$_{SNO}$ . In practice, we see only small variation of the ratios; R$_{SK}$ ∼ 1.76 - 1.8, and R$_{SNO}$ ∼ 2.2 - 2.5. These characteristics in the parameter dependence of R$_{SK}$ and R$_{SNO}$ are consistent with the result of ref. [32]. The parameter dependence which we confirmed in R$_{SK}$ and R$_{SNO}$ as discussed above could be an important observational signature to discriminate neutrino mass hierarchy, normal or inverted, as well as the mixing angle, θ$_{13}$ . In normal hierarchy, the ratios R$_{SK}$ and R$_{SNO}$ are small regardless of the mixing angle θ$_{13}$ and the shock effect. Furthermore, R$_{SNO}$ changes a little according to the value of θ$_{13}$ and R$_{SK}$ hardly changes. We defined R$_{x}$ as the total events of all flavors. Most of the events detected in SK is ¯ ν$_{e}$ . As a result, the variation of ν$_{e}$ events is hindered by larger ¯ ν$_{e}$ events unchanged by the θ$_{13}$ variation. On the other hand, the events of SNO contain the events of ν$_{e}$ as much as that of ¯ ν$_{e}$ , because the cross section of ν$_{e}$ -induced reaction is of the same order as that of ¯ ν$_{e}$ -induced reaction. Therefore, we can see the variation of ν$_{e}$ . However, the variation of ν$_{e}$ is small because the number of events is not so different between ν$_{e}$ and ¯ ν$_{e}$ . In inverted hierarchy, the ratios R$_{SK}$ and R$_{SNO}$ correlate with the mixing angle θ$_{13}$ and the shock effect. Larger θ$_{13}$ value indicates larger R$_{SK}$ and R$_{SNO}$ ratios. However, the shock effect reduces R$_{SK}$ and R$_{SNO}$ even in large θ$_{13}$ value. Therefore, we do not distinguish the value of θ$_{13}$ and the shock effect in R$_{SK}$ and R$_{SNO}$ . In the previous subsection, we discussed the ratio of high-energy to low-energy events, R$_{x}$ , which is the ratio of the neutrino events integrated over the time. We also evaluated in Sec. III C the time-dependent event rates integrated over the neutrino energy. Figures 6 and 7 indicate that some effects of shock wave propagation could be observed at later times when the shock wave reaches H-resonance. However, we apparently lose some important information on the effects of shock wave in either case. In this subsection, we explore for the signature of the shock wave effects by taking account of double differential as to both 2. Time-dependent ratio of neutrino events 17