  The relativistic Cowling approximation, where the metric perturbations are
neglected during the fluid oscillations, is often adopted for considering the
gravitational waves from the protoneutron stars (PNSs) provided via
core-collapse supernova explosions. In this study, we evaluate how the Cowling
approximation works well by comparing the frequencies with the Cowling
approximation to those without the approximation. Then, we find that the
behavior of the frequencies with the approximation is qualitatively the same
way as that without the approximation, where the frequencies with the
approximation can totally be determined within $\sim 20\%$ accuracy. In
particular, the fundamental mode with the Cowling approximation is
overestimated. In addition, we also discuss the damping time of various
eigenmodes in gravitational waves from the PNSs, where the damping time for the
PNSs before the avoided crossing between the $f$- and $g_1$-modes, is quite
different from that for cold neutron stars, but it is more or less similar to
that for cold neutron stars in the later phase. The damping time is long enough
compared to the typical time interval of short-Fourier transformation that
often used in the analysis, and that ideally guarantees the validity of the
transformation.
