When sin $^{2}$2 θ$_{13}$ is as small as 10 − $^{5}$, we see only small differences of R$_{SK}$ ( t ) between the cases with and without the shock wave and hardly see the time dependence. The H-resonance is non-adiabatic whenever the shock wave is on the resonance or not in this case of small sin $^{2}$2 θ$_{13}$ . To conclude, we can see a clear signature of the shock wave in the time-dependent high- energy to low-energy ratio of R$_{x}$ ( t ) although the influence of the shock wave is less clearly seen in the ratio of time integrated event rates, R$_{x}$ . Therefore, the observations of the time evolution of R$_{x}$ ( t ) is quite an important observable to constrain the neutrino oscillation parameters and to find the effects of shock propagation in supernovae. Fogli et al.[27] found qualitatively similar result to ours in the calculated event rates with and without forward shock in the case of inverted hierarchy for sin $^{2}$2 θ$_{13}$ = 10 − $^{2}$. Their results are however different from ours quantitatively for several reasons. For one, their event rates at the neutrino energy 45 ± 5 MeV are calculated by assuming the next generation detector of 0.4 Mton pure water that is 12.5 bigger than 32 kton for SK. We can scale our calculated results by 12.5 times in order to remove this apparent difference. The second reason is that they adopted completely different supernova model from ours. Although it is hard to reconcile the difference between the two models, we tried to compare at least the effect of shock wave propagation in the following manner. We should, first, refer to our result of the event numbers at the same neutrino energy 45 ± 5 MeV at 5 s after the core bounce for sin $^{2}$2 θ$_{13}$ = 10 − $^{3}$. This aims to compare the two results of the event rates so that the shock wave effect appears most remarkably in both calculations in the case of inverted hierarchy. Second, we should remove the bias based on different supernova models especially arising from different density profiles from each other. For this purpose we normalize our event number without shock to Fogli’s event number without shock. We thus obtain finally the following event numbers; 135 with shock and 300 without shock in inverted hierarchy. These scaled and normalized numbers should be compared with Fogli’s results; 130 with shock and 300 without shock. Note again that the latter number in either calculation is the same by definition of normalization. Our numerical calculations show 55.0 % decrease in the event number as a resultant net effect of 19 E. Further discussion