shock, which is generated by the collapse of the iron core, until the shock wave reaches the hydrogen rich envelope. Throughout one calculation, one finds that the density of the central core reaches ∼ 10 15 g / cm 3 and the shock wave propagation for beyond the hydrogen envelope ( ∼ 1 g / cm $^{3}$) is solved simultaneously [52, 53]. Figure 1 shows the calculated density profiles as a function of radius for every 1 s from 0 to 7 s after the core bounce. The shock wave reaches the oxygen-rich layer in 1 s and passes through the helium layer in 10 s. The horizontal lines show the density at the H-resonance of the neutrino energy. The shock wave reaches the H-resonance in about 2 s. The density behind the shock wave hardly decreases because we neglected neutrino cooling of the proto- neutron star in the present study. Therefore, the shock velocity is kept almost constant at 9 × 10 8 cm/s in this model. When the neutrino cooling of the proto-neutron star is taken into consideration, the density and the pressure behind the shock would inevitably decrease. As a result, the shock wave would be decelerated. We study detailed analysis of the cooling effect of the proto-neutron star elsewhere. We assume the neutrino energy spectra with Fermi-Dirac distributions. Numerical simu- lations of supernova explosions which include neutrino transport suggested that the resultant neutrino spectra at low-to-intermediate energies do not exactly follow Fermi-Dirac distribu- tions. However, since we are interested in the event number of neutrinos to be detected in a water Cherenkov detector, the neutrino signals are sensitive to high energy neutrinos because the neutrino-induced reaction cross sections in the detector are proportional to the neutrino energy. As large as the neutrino energies above ∼ 10 MeV are concerned, therefore, Fermi- Dirac distributions with finite chemical potentials may be justified as one of the reasonable approximations to the results of numerical simulations of the neutrino transport ([54]; and references therein). Although several simulations suggest different neutrino temperatures from one another, they all satisfy a common hierarchy T$_{ν}$$_{e}$ < T$_{¯}$ ν$_{e}$ < T$_{ν}$$_{x}$ . The interactions between ν$_{e}$ and neutrons freeze out at lower temperature and density than the interactions between ¯ ν$_{e}$ and protons do inside the neutralized supernova core, while the interactions be- tween ν$_{x}$ (= ν$_{µ}$ , ¯ ν$_{µ}$ , ν$_{τ}$ , and ¯ ν$_{τ}$ ) and nucleons freeze out at much higher temperature and density because ν$_{x}$ interact with matter only through the neutral current. In the present study, the neutrino temperatures are set to be T$_{ν}$$_{e}$ = 3 . 5 MeV, T$_{¯}$ ν$_{e}$ = 4 MeV, and T$_{ν}$$_{x}$ = 7 MeV associated with the neutrino chemical potentials µ$_{ν}$$_{e}$ = 7 . 28 MeV, µ$_{¯}$ ν$_{e}$ = 10 MeV, and µ$_{ν}$$_{x}$ = 0 MeV. These parameter values are well approximated to the time integrated spectra 6